Ingredients

• 3 tablespoons unsalted butter
• 2 1/2 tablespoons all-purpose flour
• 3 cups whole milk, heated
• 1 garlic clove, smashed
• 3/4 teaspoon salt
• 1/8 teaspoon black pepper
• 2 cups boiling-hot water
• 1 ounce dried porcini mushrooms (1 cup)
• 1 cup chopped onion (1 medium)
• 2 tablespoons extra-virgin olive oil
• 2 (14- to 15-ounce) cans diced tomatoes in juice
• Pinch of sugar
• 2 tablespoons chopped fresh basil
• 1/2 teaspoon salt
• 12 long ruffle-edged dried lasagne noodles (not no-boil)
• 5 ounces finely grated Parmigiano-Reggiano (1 3/4 cups)

Instructions

1. 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.
2. Make mushroom tomato sauce: 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.
3. 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.
4. 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.
5. Assemble and bake lasagne: 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.
6. 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.
7. Put oven rack in middle position and preheat oven to 425°F.
8. 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.
9. 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.