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