Position a rack in the highest level of the oven and preheat to 400 degrees F.
Heat 2 tablespoons of the oil in a 12-inch cast-iron skillet over high heat until very hot, about 3 minutes. Sprinkle the chicken thighs with salt and pepper and add them to the pan, skin side down, and lower the heat to medium. Cook, without turning, until the skin renders its fat, becomes brown and crisp and releases easily from pan, about 12 minutes.
Move the thighs to the far edge of the pan, shingling them against the side so you can cook the vegetables. Add the carrot, onion, 2 of the garlic cloves and the pancetta and cook, stirring, until the vegetables are tender and translucent, about 6 minutes. Stir in 2 tablespoons of the breadcrumbs.
Reserve 1/2 cup beans and mash them with a fork to make a chunky paste. Stir the remaining beans, chicken broth, thyme sprigs and bay leaves into the pan and season with pepper. Stir the mashed beans into the pan. Nestle the sausages and the chicken pieces into the bean mixture, placing chicken skin-side up. Bring to a simmer, place the skillet in the oven and cook until the chicken is cooked through and the sauce is thick, bubbling and has formed a skin, about 30 minutes.
Remove the pan from the oven, and set the broiler to high.
Grate the remaining 2 garlic cloves into a medium bowl with the remaining breadcrumbs and the chopped thyme. Add the remaining 1 tablespoon olive oil and stir with a fork to evenly coat. Sprinkle the breadcrumbs evenly over the cassoulet and broil until the breadcrumbs are golden and the sauce is bubbly and has formed sort of a crust, rotating the skillet so it evenly browns, about 5 minutes. Serve the cassoulet in the skillet.
The temperature of broilers can vary greatly so watch the skillet closely. It only takes a few seconds to burn.
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