Preheat oven to 400°F. Position a rack in the center.
Warm olive oil in a large oven-safe skillet over medium heat. Soften onion, carrots, and celery for 6–8 minutes.
Stir in butter until melted, then add garlic. Cook 30 seconds until fragrant.
Sprinkle in flour and cook, stirring, for 2 minutes to form a blond roux.
Gradually whisk in chicken broth, then milk. Simmer, stirring often, until thick and creamy, 4–6 minutes.
Season with thyme, parsley, salt, pepper, paprika, and Worcestershire. Fold in chicken and peas. Heat 2 minutes.
If not using an oven-safe skillet, transfer filling to a 9×13-inch baking dish and spread evenly.
Whisk flour, baking powder, garlic powder, salt, and sugar in a bowl. Cut in cold butter until the mixture looks like coarse crumbs.
Fold in cheddar and parsley. Pour in buttermilk and stir just until a thick, sticky dough forms.
Drop heaping spoonfuls of dough over the hot filling, spacing evenly to cover most of the surface.
Bake 22–26 minutes, until biscuits are puffed and golden and the filling bubbles around the edges.
Stir together melted butter, garlic powder, and parsley. Brush over hot biscuits. Rest 5 minutes before serving.