Bring a large pot of salted water to a boil. Add the spaghetti and cook until it’s tender but still firm to the tooth, about 9 min. Reserve 1 cup of the pasta cooking water and then drain the pasta and set aside.
Meanwhile, heat the olive oil and 2 Tbs. of the butter in a 12-inch skillet over medium-high heat until the butter is melted. Add the mushrooms, season with salt and pepper, and cook, stirring occasionally, until they’re brown and tender, 4 to 5 min. Transfer the mushrooms to a bowl and set aside.
In the same sauté pan, melt the remaining 6 Tbs. butter over medium heat. Add the sage leaves, and cook, stirring occasionally until they darken and crisp and the flecks of milk solids in the butter are golden brown, 3 to 5 min. Return the mushrooms to the pan and pile in the walnuts, the cooked pasta, and 1/2 cup of the pasta water. Toss the pasta continuously with tongs to coat well, adding more water as needed so the pasta is moist, 1 to 2 min. (If your skillet isn’t big enough, you can toss everything together in the pasta pot.) Season with salt and pepper, mound into bowls, and sprinkle generously with the Parmigiano. Serve immediately.