Step 1Trim the top and bottom from the squash; peel it and halve lengthwise. Discard the seeds and strings, and then cut into 1/2-inch cubes. Over low heat, melt 1/4 cup (1/2 stick) of butter in a large, deep skillet. Add 2 tablespoons of minced fresh sage, 1 tablespoon of thyme leaves and some ground pepper. Cook less than a minute, just until fragrant. Add the squash plus 2 1/2 cups of chicken broth. Bring to a boil, cover and simmer until the squash is tender, about 10 minutes. Remove with a slotted spoon to a serving dish.
Step 2Set aside 2 1/2 cups of squash plus 3/4 cup of the liquid for the pasta recipe. Reduce the remaining liquid in the pan until it is syrupy and then pour it over the squash and serve garnished with a few tablespoons of grated Parmesan.
Step 3Bring a large pot of salted water to a boil for the pasta. Cook the pasta until al dente, about 10 to 12 minutes.
Step 4While the pasta is cooking, melt the butter in a large skillet over medium heat. Add the onion and cook until softened but not brown, about 4 minutes. Add the sage and cook 1 minute longer. Stir in the squash, braising liquid and parsley. Cook just until heated through. Add salt and pepper to taste.
Step 5Drain the pasta and place it in a warm serving bowl. Add the squash mixture from the skillet, plus the Parmesan, and toss gently to combine. Serve immediately.