Preheat oven to 350F. butter a 2-quart baking dish and set aside. Bring a medium saucepan full of water to a boil. Sprinkle water with thyme. Sprinkle chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and add to water; reduce heat, cover and simmer until cooked through, about 15 minutes. Remove from water and place on cutting board to cool. Toss cooking water. {Skip this step if using rotisserie chicken}

While chicken is cooking, melt 1 tablespoon butter in a large saucepan. Add mushrooms and onions and cook until just tender, about 3-5 minutes, stirring constantly. Remove from pan and set aside.

Melt remaining 4 tablespoons butter in same saucepan and reduce heat to medium-low. Whisk in flour until well combined. Stir in chicken broth, milk, 1 tablespoon thyme, and remaining 1/2 teaspoon salt and 1/4 teaspoon pepper. Bring to a boil and then reduce heat, cooking and stirring until thickened. Remove 1 cup of the sauce and set aside.

Add in carrots and peas into the pan and stir into sauce. Cover and cook until tender, 8-10 minutes, stirring occasionally.

Coarsely chop chicken and add to sauce. Add in mushroom mixture and stir to combine. Simmer until heated through.

Spread 1/3 cup of the reserved sauce in prepared baking dish. Top with three noodles, half the chicken mixture and a third of the cheese. Repeat layers, ending with 2/3 cup of sauce {do not add final layer of cheese yet!}.

Bake, covered, for 40 minutes. Uncover and sprinkle with remaining cheese. Bake for an additional 10 minutes or until noodles are tender and cheese is browned and bubbly.