Cheesy Chicken and Mushroom Lasagne

Name: Cheesy Chicken and Mushroom Lasagne
Ingredients: 1 (10-ounce) package cremini or white mushrooms, thinly sliced; 3 garlic cloves, minced; 1 tablespoon olive oil; 5 tablespoons unsalted butter, divided; 1/2 cup dry white wine; 1/2 roast chicken, skin discarded, meat shredded (about 2 1/4 cups), and carcass reserved for stock; 3 1/2 cups whole milk; 1/4 cup all-purpose flour; 2 teaspoons thyme leaves; 3/4 cup grated Parmigiano-Reggiano; 12 Barilla no-boil egg lasagne noodles (less than a 9-ounce package); 1 1/2 cups coarsely grated Gruyère (3 ounces); Equipment: an 8-inch square baking pan