1. Preheat oven to 425 degrees. In a small bowl, mash goat cheese with a fork until smooth. Stir in pepperoncini and thyme and season with salt and pepper. With a paring knife, cut a 1-inch horizontal slit in the thick end of each chicken breast. Using your fingers, create a deep pocket between meat and bone. Tightly pack each pocket with about 2 tablespoons cheese mixture.
2. Heat a large cast-iron or other heavy, oven-proof skillet over medium-high heat. Pat chicken dry with paper towels, season with salt and pepper, and arrange in skillet, skin side down. Cook 5 minutes. Transfer skillet to oven and bake until chicken is cooked through and skin is golden and crisp, about 18 to 20 minutes. Flip chicken over in skillet and let rest 3 minutes before serving.
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