Place the dried mushrooms in a small saucepan and cover with about 1 1/2 to 2 cups water. Bring to a boil, then lower the heat and simmer for 10 minutes to reconstitute the mushrooms and flavor the water. Remove the mushrooms with a slotted spoon and finely chop. Strain the liquid through a fine-mesh strainer. Bring a large pot of water to a boil, salt it, add the 2 pastas and cook until al dente. Drain, reserving 1/2 cup of the pasta cooking water.
While the pasta is working, in a large skillet or dutch oven, heat the EVOO, 4 turns of the pan, over medium-high heat. Add the portobellos (or creminis) and brown for 10 minutes. Stir in the garlic and thyme for 2 to 3 minutes; season with salt and pepper. Deglaze the pan with the marsala and cook off the alcohol. Add the greens to wilt and season with nutmeg. Add the reconstituted mushrooms and 1 cup of their strained broth.
Add the butter to melt. Add the pasta and toss for 1 minute to combine the flavors, adding the reserved pasta cooking water if necessary. Season to taste and serve in bowls, topped with the cheese.