Preheat the oven to 425 degrees. In a large pot of boiling salted water, cook the lasagna noodles until almost tender, about 5 minutes. Drain and transfer the noodles to a bowl of cold water and let stand for 2 minutes, then drain. Pat the noodles dry. Transfer to a bowl and toss with olive oil.

In a medium skillet, heat 1 tablespoon of the olive oil. Add the Italian sausage, cover and cook over moderate heat, turning once, until browned all over. Add the water, cover and simmer until the sausage is just cooked through, about 4 minutes.

In a large skillet, heat the remaining 2 tablespoons of olive oil. Add the garlic and cook over low heat until golden, about 3 minutes. Add the tomatoes with their juices and cook over moderate heat for 10 minutes, stirring occasionally. Add the sausage and its poaching liquid and simmer for 4 minutes. Transfer the sausage to a plate. Simmer the sauce over moderate heat until thickened, about 12 minutes. Coarsely break up the sausage and season the sauce with salt and pepper.

In a well-buttered, 9-by-13-inch ceramic baking dish, arrange 3 lasagna noodles in different directions in the dish, leaving about 2 inches of overhang. Spoon a scant 1/4 cup of the tomato sauce over each lasagna noodle and sprinkle with a little grated Parmigiano-Reggiano cheese. Set a piece of mozzarella and Fontina on each lasagna noodle and add a few chunks of sausage. Fold the overhanging lasagna noodles on top of the cheese and sausage. Repeat the process with the remaining lasagna noodles, tomato sauce, mozzarella, Fontina and sausage, sprinkling with a little more Parmigiano-Reggiano cheese. Brush the softened butter on any bare pasta and curly edges and sprinkle with Parmigiano-Reggiano.

Bake the lasagna on the top rack of the oven for 20 minutes, until the sauce starts to bubble. Raise the oven temperature to 450 degrees and bake for about 7 minutes longer, until the top is richly browned. Let the lasagna rest for about 10 minutes, then scatter the sliced basil on top, cut into squares and serve.

The unbaked lasagna can be refrigerated overnight.