Heat oven to 400°F. Grease a 9x13-inch baking dish.
Warm olive oil and 1 tbsp butter in a large skillet over medium heat.
Sauté onion, carrots, and celery until slightly tender, 5–6 minutes. Stir in garlic for 30 seconds.
Melt remaining butter in the pan. Sprinkle in flour and cook, stirring, 1–2 minutes to form a roux.
Slowly whisk in warm broth, then milk. Simmer, stirring, until thick and silky, 3–5 minutes.
Fold in chicken, peas, thyme, salt, pepper, and half the parsley. Taste and adjust seasoning.
Spread the filling in the dish. Sprinkle cheddar over the top if using.
Arrange biscuits evenly over the filling. Split large biscuits in half for more coverage.
Bake until biscuits are deep golden and the filling bubbles, 22–28 minutes. Rest 10 minutes.
Finish with remaining parsley and serve warm.