1⁄2 cup lightly packed, freshly grated Parmigiano-Reggiano cheese;
1 cup finely chopped prosciutto (about 4 ounces);
Kosher salt and ground black pepper, to taste.
8 ounces fillet beans, stem ends trimmed and cut into 2-inch lengths (about 2 cups);
11⁄2 pounds waxy potatoes, such as Yukon Gold, peeled and sliced 1⁄8-inch thick (about 6 cups);
12 no-boil lasagna noodles (about 8 ounces);
11⁄2 cups freshly grated Gruyère cheese;
1⁄2 cup freshly grated Parmigiano-Reggiano cheese;
Vegetable oil spray.
For the sauce:
Bring the milk, onion, garlic, and thyme to a simmer in a small saucepan over medium heat. Remove from the heat and set aside to steep for 1 hour. Strain though a mesh sieve into a large bowl, pressing on the solids with a spoon. Discard the solids. Return the milk to the saucepan and keep warm over low heat.
Melt the butter in a medium saucepan over medium heat. Whisk in the flour and cook, whisking, for 2 minutes. Do not let the flour brown, so adjust the heat as needed. Slowly pour in the warm milk, whisking continuously. Cook, stirring slowly and steadily with a heat-proof rubber spatula, until the sauce is smooth and slightly thickened, about 3 minutes. Remove from the heat and stir in the cheese and prosciutto. Season with salt and pepper. Carefully press a piece of plastic wrap directly onto the surface of the sauce to prevent a skin from forming. Set aside.
For the lasagna:
Preheat the oven to 350 F. Butter a 9 × 13-inch glass or ceramic baking dish.
Bring a medium saucepan of water to a boil. Add 1⁄2 teaspoon kosher salt per cup of water. Add the beans and cook until just tender, about 5 minutes. Drain in a colander and rinse under cold running water until cool. Drain and pat dry.
Place the potatoes in a large saucepan and cover with cold water. Add 1⁄2 teaspoon kosher salt per cup of water. Bring to a boil, reduce the heat, and simmer until almost tender, about 8 minutes. Drain well and pat dry.
Remove the plastic wrap from the cream sauce and scrape any sauce stuck to the plastic back into the pan. Warm over low heat. Spread 1⁄4 cup of the sauce over the bottom of the prepared baking dish. Cover with 4 lasagna noodles, overlapping the edges slightly as needed. Drizzle 3⁄4 cup of sauce over the noodles. Layer in half of the beans, half of the potatoes, and half of the Gruyere. Cover with 4 lasagna noodles and drizzle with 3⁄4 cup of sauce. Layer in the rest of the beans, the potatoes, and the Gruyere. Top with the remaining 4 lasagna noodles. Pour the rest of the sauce over the top. Sprinkle with the Parmigiano-Reggiano.
Mist the shiny side of a sheet of aluminum foil with vegetable oil spray and cover the dish, sprayed side down. Bake for 20 minutes, remove the foil, and continue baking until the top is golden and the edges are bubbling, about 25 minutes more.
Let stand for at least 10 minutes before serving hot.
What else works? You can use asparagus spears in place of the fillet beans.