Heat the oven to 400ºF. Spread the squash pieces and garlic on a rimmed baking sheet and sprinkle them with 2 tablespoons of the oil, ¼ teaspoon of the salt, and ⅛ teaspoon of the pepper. Roast until the squash is tender and lightly browned, 20 to 25 minutes.
Remove the garlic to a cutting board and let it cool. Squeeze the cooled garlic from the skins (it should be a soft paste).
Meanwhile, bring a large pot of salted water to a boil. Stir in the pasta, and cover the pot to quickly return the water to a boil. Cook the pasta until tender yet chewy, 10 to 12 minutes.
While the pasta cooks, toast the walnuts in a large deep skillet over medium heat until fragrant, 1 to 2 minutes, shaking the pan now and then. Remove the nuts to a cutting board to cool; then chop them.
Add the butter, sage, garlic paste, ½ cup of the pasta water, and remaining 3 tablespoons of oil to the skillet and simmer the mixture over medium heat for 3 minutes. Using a skimmer, drain the pasta by transferring it to the skillet. Add another ½ cup of pasta water, and simmer and stir for 1 minute.
Remove the skillet from the heat and stir in the cheese and remaining ¾ teaspoon of salt and ⅛ teaspoon of pepper until the liquid is slightly creamy and clings to the pasta. Fold in the squash, divide the pasta among plates, and top with the toasted walnuts.
Tip: To make butternut squash easy to peel, prick the skin all over with a fork and then microwave on high for 1 to 2 minutes. Cool the squash slightly; then use a vegetable peeler to remove the skin.