Place a rack in the lower third of the oven; preheat to 450°.

Whisk together anchovies, garlic, 3 tablespoons mayonnaise, 1 tablespoon mustard, 1 tablespoon oil, and 1½ teaspoons pepper in a small bowl.

Set aside 1 tablespoon of the anchovy mayo in another small bowl.

Pat chicken dry; season outside and inside all over with salt.

Arrange breast side up in a cast-iron skillet and tuck wings underneath.

Arrange shallots around (if using legs, nestle under and around); season with salt and pepper.

Brush remaining anchovy mayo all over the chicken, making sure to get it into the nooks and crannies, then brush the shallots with any leftover anchovy mayo.

Place the chicken in the oven so legs are facing toward the back (this is the hottest part of the oven and will help the legs cook before the breast dries out) and roast until some anchovy mayo and fat begin to drip onto shallots, about 15 minutes.

Remove from oven and, using tongs, turn shallots to coat in drippings.

Return skillet to oven and continue to roast chicken, stirring shallots once or twice, until golden brown and an instant-read thermometer inserted into the thickest part of breast registers 155°, 45–55 minutes. If skin starts to get too dark on the top before the chicken is done, tent the area with foil, leaving the rest of the bird exposed. If using chicken legs, start checking at 40 minutes (a thermometer inserted right at the joint should register 160°).

Transfer the chicken and shallots to a cutting board, leaving behind any juices and fat in the skillet.

If shallots need more time to soften and darken, roast a bit longer without chicken before proceeding. Reserve skillet.

Reduce oven temperature to 400°.

Finely grate half of the zest of 1 lemon into a large bowl.

Cut the lemon in half and squeeze in the juice.

Add the reserved 1 tablespoon of anchovy mayo, the remaining 3 tablespoons of mayo, and the remaining 1 tablespoon of oil and whisk to combine.

Stir in Parmesan.

Season with salt and pepper. Set dressing aside.

Add bread to the reserved skillet along with the fat and turn to coat.

Return the skillet to the oven and toast the bread, tossing halfway through, until golden brown and crisp, 12–15 minutes.

Transfer the croutons to the bowl with the reserved dressing.

Add romaine and gently toss until lettuce is well coated.

Season salad with salt and pepper.

Slice the remaining lemon into wedges.

Carve the chicken and nestle it back into the skillet or transfer to a platter.

Arrange the shallots and lemon wedges around.

Serve with salad and more mustard alongside.