In a medium saucepan, melt 3 Tbs. 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, 3-5 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 Tbs, butter over low heat. Add the onions and chopped garlic and cook until softened, about 5 minutes.
Stir in the mushrooms, increase heat to medium-high and cook, stirring often, until dry, 5-7 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 2 quart rectangular baking dish, and spread 3/4 cup reserved garlic sauce on the bottom. Drain the lasagna noodles and lay 3, lengthwise, in the baking dish, trimming the ends with scissors to fit the pan.
Top with half of the mushroom mixture, half the mozarella cheese and a layer of noodles and repeat.
Combine the remaining garlic sauce and basil cream; spoon on top. Sprinkle with the remaining 3/4 cup parmesan.
Bake until golden, 40-45 minutes. Let stand 15 minutes.