In a large saucepan, melt 2 tablespoons butter over medium-high heat. Add the sausage and cook, breaking up the meat, until no longer pink, about 7 minutes. Using a slotted spoon, transfer to a bowl. Add the smashed garlic to the pot and cook, stirring, for 2 minutes. Add the tomatoes and bring to a boil over medium heat. Cook, stirring occasionally, for 1 hour; season with pepper.
Meanwhile, heat the milk in a microwavable measuring cup for 90 seconds. In a medium saucepan, melt 4 tablespoons butter over medium heat. Add the chopped garlic and cook for 1 minute. Whisk in the flour, then gradually whisk in the warm milk and bring to a simmer. Cook, whisking, until thickened, about 5 minutes. Stir in 2/3 cup parmigiano-reggiano and season with salt and pepper.
In a large pot of boiling, salted water, combine the remaining 1 tablespoon butter and the lasagna noodles. Cook, stirring often, until al dente; drain and rinse with cold water.
Meanwhile, in a bowl, combine the remaining 1/3 cup parmigiano-reggiano, the ricotta, eggs, basil and 1/4 teaspoon pepper.
Preheat the oven to 375 degrees F. Grease a 10-by-15-inch baking dish and a 10-by-15-inch piece of parchment paper. Coat the bottom of the dish with tomato sauce. Top with a slightly overlapping crosswise layer of lasagna noodles. Layer with half of the cooked sausage, 1 1/2 cups tomato sauce, half of the ricotta mixture and one-third of the mozzarella. Arrange another single layer of noodles crosswise in the dish. Spread evenly with the white sauce. Layer with more noodles, the remaining sausage, 1 1/2 cups tomato sauce, the remaining ricotta mixture, half of the remaining mozzarella, a final layer of noodles and the remaining tomato sauce. Top with the prepared parchment, greased side down. Cover with foil, place on a baking sheet and bake for 1 hour. Discard the foil and parchment; top with the remaining mozzarella and bake for 15 minutes. Let stand before serving.