Special equipment: an 18-by-13-inch sheet pan
Position a rack in the center of the oven and preheat to 450 degrees F.
Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the sausage and cook, breaking up the meat with a wooden spoon, until browned and cooked through, about 5 minutes. Transfer the sausage to a bowl with a slotted spoon and reserve the drippings in the pan.
Reduce the heat to medium. Add the remaining 2 tablespoons olive oil and the garlic to the pan. When the garlic begins to brown at the edges, about 1 minute, add the tomatoes with their juices, 3/4 teaspoon salt and 1/4 teaspoon pepper. Bring to a simmer and continue to cook, stirring occasionally, until the sauce thickens, about 8 minutes.
Mix the ricotta, egg, milk, basil, 1/2 cup of the grated Parmesan, 3/4 teaspoon salt and 1/4 teaspoon pepper together in a large bowl. Spread 2/3 cup of the tomato sauce across the bottom of an 18-by-13-inch sheet pan. Next, space out 6 lasagna noodles on top of the sauce, turning them to fit so there are 2 noodles across and 3 noodles down. (The noodles will expand when they cook.) Top the noodles with all of the ricotta mixture, all of the sausage, 2/3 cup tomato sauce, and half the shredded mozzarella. Add another layer of 6 lasagna noodles and top with the remaining sauce, mozzarella, and 1/4 cup grated Parmesan.
Lightly oil one side of a large piece of foil. Cover the pan with the foil, greased side down. Bake until the noodles are tender when pierced with a paring knife, about 20 minutes. Carefully remove the foil and continue to bake until the top is very crusty and brown, 10 to 15 minutes more.
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