01 - Preheat oven to 400°F. In a large ovenproof skillet or Dutch oven, melt the butter over medium heat. Sauté the onion and garlic until fragrant, about 2 minutes.
02 - Add carrots and cook for 3 minutes. Sprinkle flour over the vegetables; stir well to coat and cook for 1 minute.
03 - Gradually whisk in chicken broth and milk, scraping up any browned bits from the bottom. Bring mixture to a simmer and cook until slightly thickened, about 3 to 4 minutes.
04 - Stir in shredded chicken, peas, corn, thyme, rosemary, salt, and black pepper. Remove skillet from heat.
05 - In a mixing bowl, whisk together all-purpose flour, baking powder, salt, and garlic powder. Stir in the shredded cheddar cheese.
06 - In a separate bowl, combine buttermilk, melted butter, and lightly beaten egg. Add wet ingredients to dry; stir gently until just combined. Do not overmix.
07 - Drop generous spoonfuls of biscuit dough evenly over the chicken mixture in the skillet.
08 - Transfer skillet to oven and bake uncovered for 25 to 30 minutes, until biscuits are golden brown and filling bubbles at the edges.
09 - Remove from oven and let rest 5 minutes before serving hot.