01 - Preheat the oven to 400°F.
02 - In a large Dutch oven or heavy pot, melt butter with olive oil over medium heat. Add diced onion, carrots, and celery; sauté until softened, about 5 to 6 minutes.
03 - Stir in minced garlic and cook for 1 minute. Sprinkle flour over the vegetables and cook, stirring constantly, for 2 minutes.
04 - Gradually whisk in chicken broth followed by whole milk, stirring until smooth. Bring the mixture to a simmer.
05 - Add shredded chicken, frozen peas, frozen corn, dried thyme, dried rosemary, salt, and black pepper. Simmer gently for 8 to 10 minutes until vegetables are tender and soup thickens. Stir in fresh parsley.
06 - While soup simmers, whisk together flour, baking powder, baking soda, and salt in a large bowl. Cut in the cold butter using a pastry cutter or fingers until the mixture resembles coarse crumbs. Stir in cold buttermilk and optional chives just until the dough comes together.
07 - Turn dough onto a floured surface, pat to 1-inch thickness, and cut into rounds using a biscuit cutter or glass.
08 - Pour the hot soup into an oven-safe casserole dish or leave it in the oven-safe Dutch oven. Arrange biscuit rounds evenly over the surface.
09 - Brush the tops of the biscuits lightly with buttermilk.
10 - Bake uncovered for 20 to 25 minutes, until biscuits are golden brown and cooked through.
11 - Allow to cool slightly before serving.