In a medium saucepan, melt 3T butter over low heat. Stir in the flour and cook for 2 minutes. Slowly whisk in the hot milk, then the smashed garlic and 3/4t salt. Bring to a boil and cook, whisking, until thickened, 5 minutes.
In large pot of boiling salted water, cook the lasagna noodles for 8 minutes. Drain and transfer to a bowl of cold water.
Preheat the oven to 375 degrees. In a skillet, melt the remaining butter over low heat. Add the onions and chopped garlic and cook for 5 minutes. Stir in the mushrooms and cook until softened, about 7 minutes.
Using a food processor, puree the basil with the cream. Set aside 1/2c basil cream. Stir the remaining basil cream and 1/2c cheese into the mushroom mixture; season with salt and pepper.
Grease a 9 x13inch baking dish; spread 1c garlic sauce on the bottom. Drain the lasagna noodles, then layer 3 in the dish. Top with half of the mushroom mixture, a layer of noodles, the remaining mushroom mixture and noodles. Spoon the remaining garlic sauce and basil cream on top. Sprinkle with the remaining 3/4c cheese. Bake until golden, 40-45 minutes.