1 1/2 pounds boneless, skinless chicken breasts
Salt and pepper
3 tablespoons vegetable oil
1 green bell pepper, chopped fine
1 poblano or Anaheim chile, seeded and chopped fine
1 jalapeno, seeded and chopped fine
1 onion, chopped fine
¼ teaspoon dried thyme
1 ½ cups (7 ½ ounces) plus 3 tablespoons all-purpose flour
2 ¾ cups chicken broth
1 cup frozen corn, thawed
2 ounces sharp cheddar cheese, shredded (1/2 cup)
3 scallions, sliced thin
1 teaspoon baking powder
⅛ teaspoon cayenne pepper
¾ cup heavy cream
Adjust oven rack to middle position and heat oven to 450 degrees. Pat chicken dry with paper towels and season with salt and pepper. Heat 12-inch cast-iron or nonstick skillet over medium heat for 5 minutes. Add 2 tablespoons oil and heat until shimmering. Brown chicken, about 4 minutes per side; transfer to plate.
Heat remaining 1 tablespoon oil in now-empty skillet over medium heat until shimmering. Add peppers 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 tablespoons 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 simmer.
Nestle chicken into skillet along with any accumulated juices. Reduce heat to gentle simmer, cover, and cook until chicken registers 160 degrees, 15 to 20 minutes, flipping chicken halfway through cooking.
Transfer chicken to carving board, let cool slightly, then chop into small chunks. Stir shredded chicken, and corn into skillet. If you used a nonstick skillet, transfer the mixture to a greased baking dish.
Whisk remaining 1 ½ cups flour, cheddar, scallions, baking powder, ¾ teaspoon salt, ½ teaspoon pepper, and cayenne together in large bowl. Stir in cream and remaining ¼ cup broth until just combined. Using greased ¼-cup dry measure, scoop out and drop 9 mounds of dough onto chicken mixture, spaced about ½ inch apart. Transfer cast iron skillet or baking dish to oven and bake until biscuits are golden brown and filling is bubbling, about 20 minutes. Let casserole cool for 10 minutes before serving.