Preheat oven to 350 degrees. Spray a 9 x 13 baking dish with non-stick cooking spray.
Cook spaghetti according to package directions. Drain and rinse with cold water. Set aside.
In a large skillet, melt butter and add mushrooms, onion and garlic. Saute for 5 minutes and add red pepper. Sprinkle flour over mixture and cook for 1 minute. Gradually stir in broth, cream and milk. Simmer until thickened.
Remove from heat and add cheeses. Stir until melted. Add salt, pepper, thyme and chicken. Stir and toss with spaghetti. Pour into prepared baking dish.
Bake for 30 minutes. Let stand 10 minutes before serving.