Mushroom Lasagnette

Name: Mushroom Lasagnette
Ingredients: 1 package of fresh pasta sheets; 1 pound cremini mushrooms thinly sliced; 5 tablespoons unsalted butter; 1 cup freshly grated mozzarella cheese; 5 tablespoons all purpose flour; 3/4 cup freshly grated Parmigiano-Reggiano; 1 large shallot finely chopped; 4 1/2 cups milk simmering (plus more if needed to thin out the sauce); 1/4 cup fresh Italian parsley finely chopped; Salt and pepper to taste