Start by roasting the butternut squash (cut in half, drizzle olive oil, roast center side down.) If you choose to boil it, peel and cut the squash in small cubes and cook for 15-20 minutes.
Bring 6 quarts generously salted water to a boil.
Once butternut squash is cooked, puree the pulp with milk and a full dash of salt. Reserve.
Cook the pasta uncover according to package time. Approximately 3 minutes before pasta cooking time is complete start making the sauce.
Melt the butter in a large, deep skillet over medium-high heat. Continue to cook butter until browned. Add garlic; saute until fragrant and sizzling, about 2 minutes. Lower the heat.
Add the puree until into the garlic butter. Grate fresh nutmeg over sauce, add pepper and more salt if needed. Reduce heat to low; stir until hot but not boiling. Keep warm over low heat. Add cheese and mix slightly.
Using tongues, transfer the pasta directly from the boiling pot to the sauce skillet. Toss and twirl and lift and toss and twirl some more until the cheese melts and the sauce coats the pasta. Use pasta water to thin out the sauce if needed.