Cook squash: Fill a medium pot 3/4 of the way with water and bring to a boil. Meanwhile, peel the butternut squash with a vegetable peeler and scrape the seeds out with a spoon. Chop squash into one-inch pieces and transfer into pot. Boil the squash until tender, about 12-14 minutes. Drain squash into a colander.
Cook pasta: Refill the pot with water and bring back to a boil to cook the pasta. Drain pasta once it's just short of al dente.
Make sauce: While the pasta is cooking, transfer the cooked squash to a blender and add milk, paprika, salt, and pepper. Blend until a smooth sauce forms. Add sauce back to the now-empty pasta pot and heat on medium heat until warm. Add cheese and stir until melted. Add noodles, stir, and serve immediately. If sauce becomes too thick, add another splash of milk.
Store leftovers in airtight container in refrigerator for up to 3 days.