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1
Preheat the oven to 450 degrees F (230 degrees C).
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2
Toss asparagus, mushrooms, and 2 tablespoons olive oil in a bowl to coat. Transfer to a baking sheet.
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3
Bake in the preheated oven until browned, about 10 minutes. Reduce oven heat to 350 degrees F (175 degrees C).
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4
Melt butter in a saucepan over medium heat. Stir in milk, salt, black pepper, and cloves. Bring to a boil, whisking constantly, until thickened, about 2 minutes. Remove sauce from heat and whisk in water.
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5
Combine remaining olive oil, onions, and garlic in a large skillet over medium heat. Cook and stir until tender, about 3 minutes. Add roasted asparagus and mushrooms.
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6
Coat a large, deep baking dish with cooking spray. Layer noodles, 1/4 of the sauce, 1/3 of the asparagus mixture, 1/2 cup mozzarella cheese, 1/2 cup Cheddar cheese, and 1/4 cup Parmesan cheese. Repeat layers twice more, followed by a top layer of noodles, 1/4 of the sauce, 1/2 cup mozzarella cheese, 1/2 cup Cheddar cheese, and 1/4 cup Parmesan cheese. Cover the baking pan with aluminum foil.
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7
Bake in the hot oven for 40 minutes. Uncover and bake until heated through, 10 to 15 minutes more.
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8
Set oven rack about 6 inches from the heat source and turn the oven's broiler to high. Broil the lasagna until brown and crisp on top, 2 to 3 minutes. Remove from the oven and let stand for 10 to 15 minutes before serving.
