Marvelously creamy-mushroomy, this lasagna gets "meatiness" from dried porcini mushrooms. Its ragù may be made a day in advance, covered, and refrigerated.
Preheat the oven to 450F. Make the ragù: Set aside a handful of sliced mushrooms. On a large rimmed baking sheet, toss the remaining sliced mushrooms with 3 tablespoons of the olive oil and 1⁄4 teaspoon salt. Roast, stirring occasionally, until cooked through and browned in some places, about 30 minutes. Set aside.
In a small bowl, combine the porcini mushrooms and boiling water and let sit for 20 minutes. Using a slotted spoon, remove the mushrooms and squeeze the excess liquid back into the bowl. Rinse, finely mince, and place into another small bowl. Pour the mushroom soaking liquid through a sieve lined with a dampened paper towel into the bowl with the porcini. Set aside.
While the vegetables cook, roughly chop the roasted cremini mushrooms, reserving any juices. Then add to the saucepan of vegetables the cremini and porcini mushrooms and their juices, the cheese rind, wine, 2 teaspoons thyme, 2-1⁄2 teaspoons salt, and 1 cup water. Bring to a boil over high heat, then lower to a simmer and cook, stirring occasionally, until the ragù is glossy and thick, 30 to 40 minutes. Remove and discard the cheese rind, stir in the parsley and 1⁄4 teaspoon pepper, and season to taste with salt. Set aside.
Make the lasagna: Bring a large pot of well-salted water to a boil, add half of the noodles, and cook, stirring occasionally, for 4 minutes. Transfer to a colander to drain. Repeat with the remaining noodles. Drain, arrange in one layer on a baking sheet, and cover with a damp towel until needed.
Preheat the oven to 375F. Spread 1⁄2 cup mushroom ragù in a 9 by 13-inch baking dish. Top with a layer of noodles. Spread 2⁄3 cup béchamel over the noodles, then top with 1 cup ragù and 1⁄2 cup Parmigiano-Reggiano. Top with a layer of noodles. Repeat layering with 2⁄3 cup béchamel, 1 cup ragù, 1⁄2 cup Parmigiano-Reggiano, noodles, 2⁄3 cup béchamel, 1 cup ragù, and 1⁄2 cup Parmigiano-Reggiano. Top with a layer of noodles. Spread the remaining béchamel over the last noodle layer, sprinkle with the remaining 1⁄4 cup Parmigiano- Reggiano, dot with the butter, top with the reserved mushroom slices, and sprinkle with the thyme.