FOR THE MERINGUE BRACKET-STYLE MUSHROOMS:

Make templates:

• Adjust the oven racks to the upper-middle and lower-middle positions and the heat oven to 200 degrees.
• Using a pencil, draw 15 half-circles ranging from 1 to 2 inches wide on 1 sheet of parchment paper, leaving at least 1½ inches between the half-circles.
• Repeat with second sheet of parchment.
• Place parchment pencil side down on 2 rimmed baking sheets.

Whip the meringue:

• Using a stand mixer fitted with a whisk attachment, whip the egg whites on medium speed until foamy, about 1 minute.
• Add the cream of tartar and salt, increase the speed to medium-high, and whip until soft peaks form, about 1 minute.
• With the mixer running, slowly add the sugar.
• Increase the speed to high and whip until very thick and stiff peaks form, 3 to 4 minutes.

Make the mushrooms:

• Fit a pastry bag with a ¼-inch round pastry tip and fill with meringue.
• Using a half-circle as a guide, pipe concentric arcs of meringue, either smooth or frilly, until the half-circle is filled.
• Repeat with the remaining half-circles.
• Pipe a stem, about ½ inch wide and 1 inch long, onto the straight side of each half-circle.
• Moisten your fingertip with water and smooth any unwanted peaks.
• Bake the meringues for 2 hours, turn off the oven, and leave the meringues in the oven until dry and crisp, about 30 minutes. (Meringues can be stored in airtight container for up to 2 weeks; store directly after cooling.)

FOR THE CHOCOLATE CRUMBLES:

Make the dough:

• Adjust the oven rack to the middle position and heat the oven to 350 degrees.
• Line a rimmed baking sheet with parchment paper.
• Combine the butter and chocolate in a medium bowl and microwave until melted, about 1 minute, stirring halfway through microwaving.
• Add the flour, sugar, cocoa, and salt and mix until thoroughly combined and a crumbly dough forms.

Bake the crumbles:

• Crumble the dough over the prepared sheet.
• Bake until the crumbles are dry, fragrant, and starting to crisp, about 15 minutes, stirring halfway through baking.
• Transfer the sheet to a wire rack and let the crumbles cool completely (crumbles will continue to crisp as they cool). Do not turn off the oven. (Crumbles can be stored in airtight container at room temperature for up to 2 weeks.)

FOR THE FILLING:

Make the espresso cream:

• Pour 1 cup of cream into a wide bowl.
• Whisk together the espresso powder and remaining 1 cup of cream in a small saucepan and bring to a simmer over medium heat.
• Remove from the heat and cover to keep hot.

Make the caramel:

• Bring the sugar, water, and corn syrup to a boil in la arge heavy-bottomed saucepan over medium-high heat.
• Cook, without stirring, until the mixture is straw-colored, 6 to 8 minutes.
• Reduce teh heat to medium-low and continue to cook, swirling the saucepan occasionally, until the mixture is a deep coppery brown and just starting to smoke, 4 to 7 minutes longer.
• Off heat, carefully whisk in the hot cream mixture a little at a time (teh caramel will bubble and steam).

Incorporate the cream cheese:

• Add the cream cheese.
• Cover and let sit for 5 minutes.
• Whisk until mostly smooth (some small flecks of cream cheese are OK).
• Transfer the mixture to the bowl with the cream and stir to combine.
• Cover and refrigerate until the mixture registers 40 degrees or below, at least 3 hours or up to 4 days.

FOR THE CAKE:

Make the batter:

• Lightly grease a 18 by 13-inch rimmed baking sheet, line with parchment paper, and lightly grease the parchment.
• Whisk the flour, sugar, baking powder, and salt together in large, wide bowl.
• Whisk the egg yolks, oil, water, and vanilla into the flour mixture until a smooth batter forms.

Whip the egg whites:

• Using a stand mixer fitted with a whisk attachment, whip the egg whites and cream of tartar on medium-low speed until foamy, about 1 minute.
• Increase the speed to medium-high and whip until stiff peaks form, 1½ to 2 minutes.

Incorporate the egg whites:

• Transfer one-third of the whipped egg whites to the batter and whisk gently until the mixture is lightened.
• Using a rubber spatula, gently fold the remaining egg whites into the batter.
• Pour the batter into the prepared sheet and spread evenly.

Bake the cake:

• Firmly tap the sheet pan on the counter 3 times to remove any large air bubbles.
• Bake until the cake springs back when pressed lightly in center, 12 to 14 minutes.
• While the cake bakes, soak a clean dish towel with water and wring out thoroughly.

Shape the cake:

• Transfer the sheet to a wire rack.
• Immediately run a knife around the edge of the sheet, then carefully invert the cake onto a second wire rack.
• Carefully remove the sheet pan and parchment.
• Lay the damp towel over cake and invert the first wire rack onto towel.
• Invert cake and remove the second rack.
• Starting from the short side, gently roll the cake and towel together into jelly roll shape.
• Let the cake cool on the rack, seam side down, for 1 hour.

Prepare the ganache:

• Bring the cream to a simmer in a small saucepan over medium heat.
• Place the chocolate and corn syrup in a bowl, pour over the cream, and let stand for 1 minute.
• Whisk the mixture until smooth.
• Let cool until the mixture has consistency of pudding, about 1 hour.

Fill and shape the cake:

• Transfer the chilled filling to the bowl of a stand mixer fitted with a whisk attachment.
• Whip on high speed until the mixture is thick and fluffy and resembles buttercream frosting, 1½ to 2 minutes.
• Gently unroll the cake with the short side parallel to the counter edge (the innermost edge of the cake will remain slightly curled; do not flatten).
• Spread the filling evenly over the cake, leaving a ½-inch margin on each short side.
• Reroll the cake, leaving the towel behind as you roll.
• Wrap the cake in plastic wrap and refrigerate for at least 20 minutes or up to 2 days.

Assemble the cake:

• Arrange two 12 by 4-inch strips of parchment 1 inch apart on a serving platter.
• Unwrap the cake and place on a cutting board.
• Using a sharp chef's knife, trim a ½-inch slice from each end of log, wiping the knife clean between cuts; discard trimmings.
• To make a branch stump, cut 1 end of the cake at a 45-degree angle, starting 1½ inches from teh end of the log (shorter side of stump will be 1½ inches long).
• Transfer the larger cake piece to the platter, centering it lengthwise on the parchment.
• To attach the stump, rest the straight side of the smaller piece against the side of the log.
• Fill in the top of the space between the pieces with about 1 tablespoon of ganache.

Decorate the cake:

• Using an offset spatula, gently spread the remaining ganache over the log, leaving the cut ends exposed.
• Use the tines of a fork to make a wood-grain pattern on the surface of the ganache.
• Carefully slide the parchment from beneath the cake (hold the stump in place with your fingertip while sliding out the parchment).
• Refrigerate the cake, uncovered, to slightly set the ganache, about 20 minutes. (Cake can be covered loosely and refrigerated for up to 24 hours; let stand at room temperature for 30 minutes before serving.)

Finish and serve:

• No more than 10 minutes before serving, use a paring knife to make a small incision in the side of the log.
• Gently insert a mushroom stem into the incision until the straight side of the mushroom rests against log.
• Repeat with the desired number of remaining mushrooms.
• Scatter the chocolate crumbles and pistachios, if using, around the cake.
• Dust the cake lightly with confectioners' sugar, if using.
• To slice, dip a sharp knife in very hot water and wipe dry between cuts.
• Serve.