Preheat your oven to 375°F and bring a large pot of salted water to boil. Cook pasta until just barely al dente, about 2 minutes under package directions since the dish will finish cooking in the oven.
While pasta cooks, heat olive oil in a large oven-safe skillet over medium heat. Sauté diced onion and minced garlic until fragrant and softened, approximately 3-4 minutes, stirring occasionally.
Add cubed butternut squash to the skillet and cook for 5 minutes, allowing edges to begin softening. Pour in vegetable broth and heavy cream, stirring gently to combine all ingredients evenly throughout.
Season the mixture with dried sage, dried thyme, salt, and black pepper to taste. Drain cooked pasta and add to the skillet, tossing gently until every strand is coated with the creamy butternut sauce evenly.
Top the baked butternut pasta recipe with grated parmesan cheese and shredded mozzarella, distributing evenly across the surface. Dot with butter pieces for extra richness and golden browning.
Transfer the skillet to your preheated oven and bake for 20-25 minutes until the cheese is bubbly and golden brown on top. The butternut squash should be completely tender when pierced with a fork.
Remove from oven and let rest for 5 minutes before serving, allowing flavors to settle and sauce to thicken slightly.