8 anchovies, mashed to a paste
8 garlic cloves, finely grated
6 Tbsp. mayonnaise, divided
1 Tbsp. Dijon mustard, plus more for serving
2 Tbsp. extra-virgin olive oil, divided
1½ tsp. freshly ground black pepper, plus more
1 3½–4-lb. whole chicken or 4 chicken legs (thigh and drumstick; about 3 lb.)
8 medium shallots, unpeeled, halved lengthwise
2 lemons, divided
1 oz. Parmesan, finely grated
4 oz. country-style bread, torn into 1½" pieces
2 romaine hearts, leaves separated, torn
Place a rack in lower third of oven; preheat to 450°. Whisk anchovies, garlic, 3 Tbsp. mayonnaise, 1 Tbsp. mustard, 1 Tbsp. oil, and 1½ tsp. pepper in a small bowl. Set aside 1 Tbsp. 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 chicken, making sure to get it into the nooks and crannies, then brush shallots with any leftover anchovy mayo.
Place chicken in 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 chicken is done, tent 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 chicken and shallots to a cutting board, leaving behind any juices and fat in 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 zest of 1 lemon into a large bowl; cut lemon in half and squeeze in juice. Add reserved 1 Tbsp. anchovy mayo, remaining 3 Tbsp. mayo, and remaining 1 Tbsp. oil and whisk to combine, then stir in Parmesan. Season with salt and pepper. Set dressing aside.
Add bread to reserved skillet with fat and turn to coat. Return skillet to oven and toast bread, tossing halfway through, until golden brown and crisp, 12–15 minutes. Transfer croutons to bowl with reserved dressing. Add romaine and gently toss until lettuce is well coated. Season salad with salt and pepper.
Slice remaining lemon into wedges. Carve chicken and nestle back into skillet or transfer to a platter; arrange shallots and lemon wedges around. Serve with salad and more mustard alongside.