1. To prepare the filling: Preheat the oven to 425 degrees F. Put the chicken in a large saucepan with the squash, mushrooms, and broth. Bring to a simmer over medium-high heat and cook just until the meat is opaque, and the squash is slightly soft, 5 to 7 minutes. Use a slotted spoon to transfer the chicken, squash and mushrooms to a 2-quart ovenproof casserole dish. Return the liquid to a boil and reduce to 2 cups, about 15 minutes.

2. Melt the butter in a large saucepan over medium-high heat. Add flour and chili powder and stir until fragrant and bubbling, 2 to 3 minutes. Whisk in the reduced broth and bring to a simmer. Cook, stirring often, until the consistency of thick gravy, about 8 to 10 minutes. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper . Pour the sauce over chicken and vegetables. (The recipe can be made up to 2 days ahead at this point. Keep covered in the refrigerator and bring to room temperature before continuing.)

3. To prepare the topping: In large bowl, combine cornmeal, 1/2 cup flour, baking powder, baking soda, and 1 1/2 teaspoons salt. In another small bowl, mix the buttermilk, egg, jalapeno, and 1 tablespoon oil. Pour the liquid ingredients into the dry ingredients and stir until just combined. Drop large spoonfuls of batter, over the chicken to cover. Bake until the topping is lightly browned and the sauce is bubbly, about 20 to 25 minutes. Serve.