Preheat oven to 425F.
Preheat olive oil in a sauté pan over medium-high heat.
Season both sides of the chicken with salt and pepper and cook, skin side down first, until golden brown, about 8 minutes. Flip and cook an additional 5 to 6 minutes until golden brown. Transfer the chicken to a tray and set aside.
Discard all but 2 tablespoons of the fat in the pan and return it to medium-high heat. Add the peppers and onion and season with a pinch of salt. Cook, stirring often, until the onions are translucent. Add the mushrooms and cook until beginning to soften, about 4 minutes. Add garlic and cook until fragrant, about 60 seconds.
Add the chicken stock and cook, scraping any brown bits on the bottom of the pan, until reduced by one quarter in volume. Add the crushed tomatoes, thyme, bay leaves, olives, capers, and potatoes, and season with ¼ teaspoon black pepper. Bring liquid to a steady simmer and return the chicken to the pan, submerging it only half way.
Transfer the pan to the oven and cook 30 minutes, or until the potatoes are fork tender and the thickest part of the chicken registers 185F on a meat thermometer.
Transfer chicken to a plate and stir in the kale. Cover with a lid and let sit until the kale is wilted, about 2 to 3 minutes. Taste the sauce for seasoning and adjust as desired. Return the chicken to the pan and serve.