Heat the oven to 425°F. Place 1 peeled and quartered small yellow onion, 1 quartered medium celery stalk, and 1 peeled and quartered medium carrot in a high-speed blender or food processor. Pulse until very finely chopped.

Season 4 bone-in, skin-on chicken thighs all over with ¾ teaspoon of the kosher salt and ½ teaspoon of the black pepper. Heat 1 tablespoon olive oil in a large, oven safe, high-sided sauté pan (at least 10 inches) or Dutch oven over medium heat until shimmering. Add the chicken skin-side down. Sear undisturbed, until the skin is very crisp and deep golden-brown, 6 to 8 minutes. Flip and cook until the second side is lightly browned, 2 to 3 minutes more. Transfer to a plate skin-side up (the chicken will not be cooked through).

Add 12 to 14 ounces cut kielbasa to the pan in a single layer. Cook, turning occasionally, until browned all over, 3 to 5 minutes total. Reduce the heat if needed if starting to burn. Using tongs, transfer to the plate with the chicken.

Add the chopped vegetables and 3 smashed garlic cloves to the pan. Cook over medium heat, stirring occasionally, until the vegetables are lightly browned, about 2 minutes. Stir in 1 tablespoon tomato paste and cook until it darkens in color, about 2 minutes. Carefully pour in ⅓ cup dry vermouth and scrape up any browned bits from the bottom and sides of the pan. Simmer until the liquid is mostly evaporated, about 2 minutes.

Stir in 2 (about 15-ounce) cans white beans with their canning liquid, 3 large fresh thyme sprigs, and the remaining ½ teaspoon kosher salt and ¼ teaspoon black pepper. Simmer until the liquid thickens slightly and the beans are well-coated, 3 to 5 minutes.

Stir in the sausage pieces. Turn off the heat. Return the chicken skin-side up to the pan, nestling it into the beans.

Transfer the pan to the oven. Roast until the chicken is cooked through and the top of the liquid has a crust, about 25 minutes. Meanwhile, make the breadcrumb topping if desired.

Heat 1 tablespoon olive oil in a small skillet over medium heat until shimmering. Add ½ cup panko breadcrumbs and a pinch of kosher salt and black pepper. Cook, stirring often, until the panko browns and crisps, 2 to 3 minutes. Take the skillet off the heat.

Let the cassoulet cool for 5 to 10 minutes. Sprinkle 2 tablespoons coarsely chopped fresh parsley leaves over the cassoulet, and serve topped with the breadcrumb topping if using.