In a medium saucepan, melt 3 tablespoons butter over low heat. Stir in the flour and cook for 2 minutes. Slowly whisk in the hot milk, then the smashed garlic clove and 3/4 teaspoon salt. Bring the garlic sauce to a boil and cook, whisking, until thickened, about 5 minutes. Set aside.
In a large pot of boiling salted water, cook the lasagna noodles for 8 minutes. Drain and transfer to a bowl of cold water.
Position a rack in the center of the oven and preheat to 375 degrees . In a large skillet, melt the remaining 2 tablespoons butter over low heat. Add the onions and chopped garlic and cook until softened, about 5 minutes. Stir in the mushrooms and cook until softened, about 7 minutes.
Using a food processor, puree the basil leaves with the cream. Set aside 1/2 cup basil cream. Stir the remaining basil cream and 1/2 cup parmesan into the mushroom mixture; season with salt and pepper.
Grease a 9-by-13-inch baking dish and spread 1 cup reserved garlic sauce on the bottom. Drain the lasagna noodles and lay 3 in the baking dish. Top with half of the mushroom mixture, a layer of noodles, then the remaining mushroom mixture and noodles. Combine the remaining garlic sauce and basil cream; spoon on top. Sprinkle with the remaining 3/4 cup parmesan.
Bake until golden, 40 to 45 minutes. Let stand for 15 minutes.