1. Rinse off the mushrooms and pat dry. Cut off the thickest part of the stems and discard. Cut the mushrooms into 1-inch pieces, roughly halves or quarters, depending on the size of the mushrooms.
2. In a large, wide skillet over medium-high heat, add the olive oil. Once the oil is hot, add the mushrooms, salt, pepper, and chile flakes. Stir to coat the mushrooms evenly with the olive oil. Saute until the mushrooms become golden, 12 to 15 minutes. Add the shallots and asparagus, stir, and cook until the mushrooms are deep golden and the asparagus is tender but firm, about 8 more minutes. Stir in the garlic and cook for 30 seconds more, just to bring out the flavor of the garlic but not to brown it.
3. Add the white wine and simmer until reduced by half. Turn off the heat and set aside. Stir in the butter until melted and fully incorporated.
4. While the mushrooms are cooking, bring a large pot of water to a boil. Once the water is boiling, cook the lasagna according to the package instructions. Once the pasta is cooked to al dente, pull it out of the water and drain (do not rinse). Rub with a touch of olive oil to prevent sticking and set 1 sheet of pasta on each of four plates.
5. Divide the Taleggio and Parmesan evenly across the pasta. Spoon the mushroom-asparagus mixture over the cheeses. The heat will partially melt the cheeses. Shower the basil on top of each plate. Drizzle a touch of extra virgin olive oil and a touch more Parmesan over each. Serve.