Wild Mushroom Lasagne

Heat butter in a 2-quart heavy saucepan over moderately low heat until melted, then add flour and cook roux over low heat, whisking, 3 minutes. Add hot milk in a fast stream, whisking vigorously, and whisk in garlic and salt. Bring to a boil, whisking. Reduce heat and simmer, whisking occasionally, 10 minutes, then remove garlic and add pepper.
Pour boiling-hot water over porcini in a bowl and let stand until softened, about 20 minutes.Lift out porcini, squeezing excess liquid back into bowl, then rinse to remove any grit. Pour soaking liquid through a sieve lined with a dampened paper towel into another bowl. Chop porcini and add to soaking liquid.
Cook onion in oil in a 12-inch heavy skillet over moderate heat, stirring occasionally, until softened, 4 to 5 minutes, then stir in tomatoes with juice, sugar, porcini with soaking liquid, and 1 tablespoon basil. Simmer, stirring frequently, until tomatoes have broken down into a chunky sauce, about 25 minutes. Stir in salt and remaining tablespoon basil.
Cook lasagne noodles in a 6- to 8-quart pot of boiling salted water 8 minutes (noodles will not be cooked through), then drain and transfer to a bowl of cold water.
Put oven rack in middle position and preheat oven to 425°F.
Spread 1 cup béchamel on bottom of a buttered 13- by 9-inch baking dish, reserving remainder for last layer. Drain 3 lasagne noodles in 1 layer on a kitchen towel and arrange over béchamel in baking dish. Spread pasta in dish evenly with one third of mushroom tomato sauce and sprinkle with 1/3 cup cheese. Repeat layering of pasta, mushroom tomato sauce, and cheese twice, then cover with last 3 lasagne noodles. Spread remaining béchamel on top and sprinkle with remaining 3/4 cup cheese.
Bake, uncovered, until lasagne is bubbling and top is browned, 30 to 35 minutes. Let stand at room temperature at least 15 minutes before cutting.