Heat oven to 400°F. Lightly grease a 9x13-inch baking dish.
Warm olive oil and 2 tbsp butter in a large skillet over medium heat.
Soften onion, carrots, and celery in the skillet, stirring occasionally, 5–6 minutes.
Stir in garlic and cook until fragrant, about 30 seconds.
Melt remaining 2 tbsp butter. Sprinkle in flour and cook, stirring, 1 minute.
Whisk in broth, then milk, a little at a time. Simmer until thick and silky, 3–5 minutes.
Off heat, stir in sour cream, thyme, parsley, salt, pepper, paprika, and half the cheese.
Fold in chicken and peas until evenly distributed.
Spread the creamy filling in the prepared baking dish.
Arrange biscuit quarters over the top and sprinkle with remaining cheese.
Bake 18–22 minutes, until biscuits are deep golden and the filling bubbles at the edges.
Let the casserole rest 10 minutes before serving so the sauce sets.