Adjust oven rack to middle position and preheat to 350°F.
Sift flour into a medium bowl (if using a cup measure, spoon into the cup and level with a knife before sifting).
Whisk in baking powder.
Combine egg yolks, sugar, salt, and vanilla in the bowl of a stand mixer fitted with the whisk attachment.
Mix on low speed to moisten, then pause to scrape the bowl.
Increase to medium-high and whip until thick and roughly doubled in volume, with a clear vortex pattern left by the whisk, about 5 minutes. Do not overbeat, as this will destabilize the foam.
Reduce speed to medium-low and drizzle in oil, followed by club soda. As soon as you’ve added the last drop, shut off the mixer and remove the bowl. The batter will look thin and foamy.
Add cake flour and gently combine with a balloon whisk.
Use a flexible spatula to fold the thin batter once or twice from the bottom up.
Lightly grease a nonstick canoe pan (see headnote) with pan spray—do not do this in advance—and fill each mold three-quarters full (about 1¾ ounces).
Bake until cakes have domed above the pan and spring back to the touch, though your fingertip will leave a light impression in the pale crust, about 12 minutes.
Cool 10 minutes, then pry the cakes from the pan while still warm and flexible.
Store in an airtight container at room temperature for up to 24 hours.