Bring a large pot of water to boil for the pasta.
In a large, deep skillet, heat the olive oil over medium- high. Add the mushrooms and cook, stirring occasionally, until browned, about 6 minutes. Add the shallots, garlic and thyme; season with salt and pepper. Cook until the shallots soften, about 2 minutes. Add the peas and sherry; cover and simmer until the peas are tender, 2 to 3 minutes. Uncover and reduce the heat to medium.
Salt the boiling water, add the pasta and cook to al dente.
Add 1 cup of the starchy pasta water to the mushroom mixture, then drain the pasta. Using tongs, toss the greens into the mushrooms to wilt, then add the pasta. Add the lemon juice and season. Serve the pasta in shallow bowls with a sprinkle of the cheese.