1. Preheat oven to 425. Cut eggplants in 1 ½″ cubes and toss them with oil and salt until evenly coated. Spread eggplant in one even layer on baking sheet. Roast for 25-30 minutes or until eggplant is evening brown and soft.
2. In a medium saute pan add 1 T. olive oil over medium heat. Add prosciutto and saute until browned, about 3 minutes. Add shallots and rosemary, saute until shallots are tender. About 4 minutes. Remove from heat and save.
3. Make the Bechamel Sauce: Bring milk and broth to a simmer over medium heat. Simmer for 3 minutes and then remove from heat.
4. Melt butter in a heavy medium sauce pan over low heat. Whisk in flour, stir carefully to make sure there are no lumps. Stir for 2 minutes to thicken. Whisk in hot milk mixture and take care to make sure there are no lumps. Bring to a boil, whisking constantly. Reduce heat to low and simmer for 5 minutes, stirring frequently. The sauce will thicken and reduce. After 5 minutes remove the sauce from the heat. Quickly add the grated Gruyere, Parmesan and nutmeg. Season with salt and pepper to your taste.
5. I am a big fan of “No Boil” lasagna noodles – they save a lot of table. So, no matter what you are using, add some sauce to the bottom of a 13x9x2 inch pan. Spread sauce evenly so it coats the whole bottom. Add one layer of noodles. Add more sauce. Then put a layer of roasted eggplant and top that with the prosciutto mixture. Follow with a second layer of noodles. Repeat the sauce, eggplant, prosciutto layering pattern. Finally, top with a last layer of noodles and top with the remaining sauce.
6. Finally, at the end, sprinkle the top with grated Parmesan cheese.
7. Preheat oven to 350 and bake until the top of the lasagna is golden and the sauce is bubbling, about 45 minutes. It may need to bake for up to an hour if it was refrigerated. Let stand for 15 minutes before serving.