How to Make It
In a medium saucepan, cover the cannellini beans with water and bring to a boil. Remove from the heat, cover and let stand for 1 hour. Drain the beans.
In the same saucepan, heat 2 tablespoons of the oil. Add the onion, carrot, celery and garlic and cook over moderate heat, stirring, until softened, 5 minutes. Add the cannellini beans and chicken stock and bring to a boil. Cover and simmer over low heat, stirring occasionally, until the beans are tender, about 1 hour and 15 minutes.
Meanwhile, roast the red pepper directly over a gas flame or under a preheated broiler until charred all over. Let cool slightly, then discard the charred skin, stem and seeds. Cut the pepper into 1/2-inch dice. Add the green beans to the cannellini beans and simmer over moderate heat until tender, about 5 minutes. Stir in the roasted red pepper and season with salt and pepper.
Preheat the oven to 375°. On a rimmed baking sheet, toss the focaccia with 1 tablespoon of the oil. Bake for 7 minutes, or until crisp; let cool. Leave the oven on.
In a small skillet, heat 1/2 tablespoon of the oil. Season the squab livers with salt and pepper and cook over moderately high heat until lightly browned, about 1 minute per side. Add the shallot and cook over low heat until softened, about 3 minutes. Finely dice the livers and add them to the focaccia along with the shallot. Stir in the thyme and season with salt and pepper.
Stuff the squab with the focaccia. Rub the butter all over the squab and season with salt and pepper. In a large ovenproof skillet, heat the remaining 2 tablespoons of oil. Add the squab, breast side up. Roast the squab in the upper third of the oven for about 20 minutes, basting every 5 minutes, until an instant-read thermometer inserted in the breast registers 130° for medium-rare. Transfer the squab to a carving board and let rest for 5 minutes. Mound the stuffing from each bird onto each plate. Carve the breast halves and whole legs and transfer to the plates. Spoon the bean stew alongside and serve.