1. Melt butter in a large saucepan over medium heat. Add garlic and sauté for one minute, stirring constantly.
2. Add butternut squash and vegetable broth and stir to combine. Continue cooking until the liquid starts to simmer.
3. Reduce heat to medium-low, cover, and cook for 10-15 minutes, stirring occasionally.
4. While squash cooks, use a large pot to prepare pasta according to package instructions. Before draining, scoop out 1 cup of pasta water and set aside. Add pasta back to pot after draining.
5. When squash is tender and mashes easily with a fork, remove from heat. Let sit 5 minutes to cool slightly.
6. Transfer squash mixture to a blender. Add milk. Open the small cap on the blender lid and hold a dish towel over it while you blend until smooth. This helps steam to escape, so the lid does not blow off. If your blender is smaller, blend in batches so you are never filling it more than halfway.
7. Add squash mixture and cheese to the pot with the pasta. Stir until cheese melts and sauce thickens. Add pasta water as needed to thin out the sauce.