Cook linguine until just al dente, then drain and set aside.
Warm olive oil and butter in a skillet, then cook the onion until soft.
Add the garlic and mushrooms, cooking until browned and fragrant.
Stir in the flour and cook briefly, then slowly whisk in the broth, milk, and cream.
Season with salt, pepper, and thyme, then simmer until lightly thickened.
Fold in the chicken, peas, pasta, half the mozzarella, and half the Parmesan.
Transfer to a baking dish, top with remaining cheese and breadcrumbs, then bake until bubbly and golden.
Finish with parsley and serve warm.
