Bring a large pot of salted water to a boil. Cook the pasta according to package directions until al dente. Reserve 1/2 cup of the pasta water, then drain.
While the pasta cooks, heat 1 tbsp olive oil in a large skillet over medium heat. Add the butternut squash, salt and pepper and cook 8-10 minutes, stirring occasionally, until tender and lightly golden. Add the garlic and cook 1 minute more.
Transfer the squash to a blender with the vegetable broth and blend until smooth. Vent the lid to let steam escape.
Heat the remaining 1 tbsp olive oil in the same skillet over medium heat. Fry the sage leaves 30-45 seconds until crisp, then transfer to a paper towel. Turn off the heat.
Wipe the skillet, pour in the squash puree and set it over medium-low heat. Stir in the milk and Parmesan and simmer 2 minutes; season to taste.
Add the pasta and toss to coat, thinning with the reserved pasta water as needed. Serve topped with the crispy sage and extra Parmesan.
