Heat oven to 375°F and grease a 9x13-inch baking dish.
Boil spaghetti in salted water until just shy of al dente; drain well.
Warm 1 tbsp olive oil in a large skillet. Sauté mushrooms and onion until browned and tender, 6–8 minutes.
Stir in garlic and cook 30 seconds. Transfer the vegetables to a bowl.
Melt butter in the same skillet. Whisk in flour and cook, whisking, 1–2 minutes.
Slowly whisk in chicken broth, then half-and-half and sherry. Simmer, whisking, until thick and creamy, 3–5 minutes.
Remove from heat. Stir in sour cream, salt, pepper, thyme, 3/4 cup Parmesan, and half the mozzarella.
Fold in chicken, peas, and the sautéed vegetables. Add cooked spaghetti and toss to coat evenly.
Spread into the baking dish. Mix breadcrumbs with remaining 1 tbsp olive oil; sprinkle over top with remaining cheeses.
Bake 20–25 minutes until bubbling and golden. Rest 5 minutes, then garnish with parsley and serve.