Roast the Squash: Preheat oven to 425°F. Spread cubed butternut squash on a baking sheet. Toss with olive oil, salt, and pepper. Roast for 25–30 minutes, stirring halfway, until golden brown and caramelized at the edges.
Cook the Pasta: Meanwhile, bring a large pot of salted water to a boil. Cook pasta until just shy of al dente. Reserve 1 cup of pasta water, then drain.
Make the Sage Brown Butter: In a large skillet, melt butter over medium heat. Add sage leaves and cook until butter foams and turns golden brown, and sage leaves crisp (about 3–4 minutes). Remove sage with a slotted spoon and set aside for garnish.
Build the Sauce: Add garlic to the brown butter and sauté until fragrant (about 1 minute). Stir in cream and reserved pasta water, whisking to combine into a silky sauce. Simmer for 3 minutes, then stir in half of the Parmesan.
Combine: Toss pasta with sauce, roasted squash, and half of the mozzarella. Adjust seasoning with salt, pepper, and optional red pepper flakes.
Bake: Transfer mixture to a greased 9x13-inch baking dish. Top with remaining mozzarella and Parmesan. Bake at 375°F for 20 minutes, until bubbly and golden.
Serve: Garnish with crispy sage leaves. Serve warm with salad or bread.