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Preheat oven to 450 F. Pat chicken dry with paper towels and season with salt and pepper. Heat a 12-inch cast iron skillet over medium heat for 5 minutes. Add 2 tbsp oil and heat until simmering. Brown chicken, about 4 minutes per side; transfer to plate.
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Heat remaining 1 tbsp oil in the skillet over medium heat. Add the celery and onion and cook until softened and lightly browned, 5 to 7 minutes. Stir in thyme and cook until fragrant, about 30 seconds. Stir in 3 tbsp flour and cook for 1 minute. Slowly whisk in 2 ½ cups broth, scraping up any browned bits and smoothing out any lumps, and bring to a simmer.
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Nestle chicken into skillet along with any juices from the plate. Reduce heat to gentle simmer, cover and cook until chicken registers 175 F, 12 to 15 minutes, flipping the chicken halfway through cooking.
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Transfer chicken to craving board, let cool slightly, and then pull into large chunks using 2 spoons. Stir in shredded chicken, bell pepper and corn into skillet.
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Whisk remaining 1 ½ cups flour, cheddar cheese, scallions, baking powder, ¾ tsp salt, ½ tsp pepper and cayenne together in a large bowl. Stir in cream and remaining ¼ cup broth until just combined. Using a greased ¼ cup dry measuring cup, scoop out and drop 9 mounds of dough onto chicken mixture, spaced about ½ inch apart. Transfer skillet to oven and bake until biscuits are golden brown and filling is bubbling, about 20 minutes. Let casserole cool for 10 minutes before serving.