In a large pot of boiling salted water, cook pasta until al dente; drain, and return to pot. Set aside. While pasta is cooking, in a small bowl, soak porcini in 1 1/2 cups hot water until tender, at least 15 minutes.
Meanwhile, melt 1 tablespoon butter in a large skillet over medium heat. Add shallot; season with salt and pepper. Cook, stirring occasionally, until shallot begins to soften, 3 to 4 minutes. Add wine; cook until almost evaporated, 3 to 5 minutes.
Add button and shiitake mushrooms. Cook over medium-high heat, stirring occasionally, until mushrooms are tender and begin to brown, 6 to 8 minutes.
Add porcini and their soaking liquid (leaving any grit at the bottom of the bowl) and thyme to skillet. Cook, stirring occasionally, until liquid reduces by half, 2 to 4 minutes. Transfer porcini mixture to pot with pasta. Add Parmesan and remaining butter, and toss to combine. Season with salt and pepper, and serve with more Parmesan.
Soaking dried porcini mushrooms in hot water softens them and creates an intense broth used for the pasta's sauce. Find dried porcini in the international aisle of markets and gourmet stores.