01 - Preheat your oven to 400°F.
02 - In a large Dutch oven or heavy pot, heat butter and olive oil over medium heat. Add onion, carrots, and celery. Sauté until softened, about 6 minutes.
03 - Stir in garlic and cook for 1 minute, until fragrant.
04 - Sprinkle flour over the vegetables and stir to coat. Cook for 2 minutes, stirring frequently.
05 - Gradually whisk in the chicken broth, scraping the bottom to deglaze. Add potatoes, thyme, sage, black pepper, and salt. Bring to a simmer and cook for 10 minutes, until potatoes are just tender.
06 - Add shredded chicken, peas, and corn. Simmer for another 5 minutes.
07 - Stir in milk or half-and-half and simmer gently for 3 minutes, not boiling. Adjust salt and pepper to taste. Remove from heat and fold in parsley.
08 - For the biscuits, whisk together flour, baking powder, sugar, and salt in a bowl. Add cold butter and cut in with a pastry cutter or fingers until mixture resembles coarse crumbs.
09 - Stir in milk until just combined. Do not overmix. Gently fold in parsley, if using.
10 - Drop heaping spoonfuls of biscuit dough over the surface of the soup in the pot or transfer soup to a large oven-safe baking dish and top with biscuits.
11 - Brush biscuit tops with a little milk.
12 - Bake uncovered for 20–25 minutes, until biscuits are golden and cooked through.
13 - Let cool for 5 minutes before serving.