Arrange the cubed butternut squash on a baking sheet. Drizzle with olive oil, salt, and pepper and bake for 20-25 minutes or until fork tender.
Trim hard parts from mushrooms and discard. Wipe mushrooms clean with a damp paper towel to remove grit; cut caps lengthwise. In a large pan over medium to high heat, add 1 tbsp of olive oil. Place the mushrooms in the pan but do not overcrowd (you may have to do this in batches). Cook the mushrooms until all the moisture is gone and the mushrooms have become crispy. This may take 10-15 minutes. Set mushrooms aside.
Cook the pasta al dente and reserve 1 cup of the water before draining. In a large pan add 1 tbsp olive oil and the garlic. Cook for 1-2 minutes before add the pasta, mushrooms, butternut squash, 1 tbsp truffle oil, 1/2 cup reserved pasta water, salt and pepper to taste. Mix well and toss in the arugula right before serving. Serve with a little more freshly grated Parmesan on top.
