How to Make It
On a work surface, cut the wings off of the chickens at the second joint, leaving the drumettes attached to the breasts. Transfer the wingettes to a small bowl. Season the chickens with salt and pepper and arrange skin side up on two large rimmed baking sheets. Cover with plastic wrap and refrigerate for at least 4 hours or overnight.
Drizzle 2 tablespoons of the olive oil over the chickens and top each half with 2 cubes of the butter. Scatter the thyme, rosemary and garlic over the chickens; let them return to room temperature.
Meanwhile, preheat the oven to 400°. On a large rimmed baking sheet, toss the bread with 1/4 cup of the olive oil and season with salt and pepper. Bake for about 10 minutes, until golden but still slightly chewy. Increase the oven temperature to 425°.
In a very large ovenproof skillet, heat 1 tablespoon of the olive oil until shimmering. Add the chicken livers and cook over moderately high heat, turning occasionally, until well browned and barely pink inside, 3 to 5 minutes. Transfer the livers to a plate to cool, then cut into 1/2-inch pieces.
Heat 1 tablespoon of olive oil in the skillet until shimmering. Add the wingettes to the skillet and cook over moderately high heat until golden on the bottom, about 4 minutes. Add the fennel, scallions, sage and a generous pinch each of salt and pepper and transfer the skillet to the oven. Roast for about 30 minutes, stirring once, until the fennel is tender and the wingettes are cooked through. Let cool slightly, then remove the meat from the wingettes; discard the skin and bones. Add the meat to the vegetables.
Roast the chickens in the upper and lower thirds of the oven for 40 to 45 minutes, until an instant-read thermometer inserted into a thigh registers 165°; switch the pans halfway through roasting. Let rest for 10 minutes.
In a bowl, toss the bread with the roasted vegetables, escarole, livers, anchovies, capers, vinegar and the remaining 2 tablespoons of olive oil. Fold in 1/2 cup of the cheese shavings and season the bread salad with salt and pepper. Transfer the bread salad to a platter and scatter the remaining 1/2 cup of cheese shavings on top. Cut the chicken halves into two pieces and transfer to another platter. Serve hot with the bread salad.