Bring a large pot of water (4 to 5 quarts) to a boil over high heat
While water is heating, chop garlic, slice mushrooms and dice pumpkin
Once the water has reached a rapid boil, add salt and then the spaghetti. Cook until very al dente (about 1 minute less than recommended cooking time on package directions)
While the spaghetti is cooking, heat the olive oil in a large sauté pan over medium heat. Add the diced squash and cook for 5 minutes, stirring often. Squash should be lightly golden and soft to the fork. Add garlic and cook for a minute; then add mushrooms and cook for a few more minutes, stirring occasionally
Season with a pinch of salt (to taste) and add fresh thyme leaves
When pasta is ready, reserve ½ cup of pasta water and drain the rest. Return spaghetti to the pot, add reserved pasta water and half of the pumpkin-mushroom mixture, stirring over low heat for a minute or so, until most of the water has been absorbed by the pasta
Remove pot from heat and add the Pecorino Romano cheese, stirring until the sauce is slightly creamy and all the flavors are well incorporated
Transfer to a platter or portion onto individual plates. Top with the rest of the pumpkin-mushroom sauce, and serve immediately, adding more grated Pecorino Romano if desired
National Pasta Association Recipe