2 tablespoons plus 1 teaspoon extra-virgin olive oil, divided
Coarse salt and freshly ground pepper
2 lemons, thinly sliced, divided
6 small shallots (6 ounces), peeled and quartered lengthwise
Preheat oven to 425 degrees. Place chicken, breast side down, on a work surface. Starting at thigh end, cut along 1 side of backbone with kitchen shears. Turn chicken around; cut along other side. Discard backbone or save for stock. Flip chicken, and open it like a book. Press firmly on breastbone to flatten.
Rub chicken with 1 tablespoon oil, and season with 1 tablespoon salt and 1/2 teaspoon pepper. Brush 1 tablespoon oil in the center of a rimmed baking sheet slightly larger than the size of the chicken, and place half the lemon slices in a single layer on top of oil. Place chicken, skin side up, on lemons. Beginning at the neck end of breast, carefully loosen skin from flesh of breast and thighs with your fingers. Slide remaining lemon slices under skin in a single layer.
Roast chicken 20 minutes. Toss shallots with remaining teaspoon oil, and scatter around chicken. Continue to roast chicken until a thermometer inserted into thickest part of breast reaches 165 degrees, 25 to 30 minutes more.
Transfer chicken to a carving board, and let rest 10 minutes. Cut chicken into 8 pieces, and serve with roasted lemons, shallots, and pan juices. If using chicken for soup and salad, let chicken cool completely. Remove and discard skin, lemon under skin, and bones, and shred meat into bite-size pieces.
Chicken, roasted lemons, shallots, and pan juices can be refrigerated up to 2 days. Store meat and pan juices in separate containers; lemons and shallots can be stored together.