Naptime Courtship Eggplant And Prosciutto Lasagna

Name: Naptime Courtship Eggplant And Prosciutto Lasagna
Ingredients: 1 lb Lasagna noodles (Use the “No Boil” to save yourself valuable time); 3 large eggplants, cubed; 2 T. olive oil; 2 T. Kosher salt; 1 ½ c. prosciutto, chopped; 2 large shallots; 1 t. chopped fresh rosemary; 4 c. whole milk; 1 ½ c. chicken broth; 1 stick unsalted butter; 2/3 c. all-purpose unbleached flour; 2 c. Gruyere, shredded; ½ c. grated Parmesan cheese (I used Pecorino-Romano once in a pinch and that tasted fine, too)