Prepare the Butternut Squash: Preheat the oven to 400°F (200°C). Toss the cubed butternut squash with olive oil, salt, and pepper. Spread evenly on a baking sheet and roast for 20–25 minutes, flipping halfway through, until tender and caramelized. Set aside.
Cook the Pasta: Bring a large pot of salted water to a boil. Add pasta and cook until al dente, according to package directions. Reserve ¼ cup of pasta water before draining.
Make the Brown Butter: In a large skillet or sauté pan, melt butter over medium heat. Once melted, add sage leaves and let them crisp gently (about 2 minutes). Remove the sage leaves and set aside for garnish. Continue cooking the butter until it turns golden brown and smells nutty — about 2–3 minutes. Stir in minced garlic and red pepper flakes, cooking for 30 seconds until fragrant.
Assemble the Pasta: Add roasted butternut squash to the brown butter sauce and toss to coat. Add cooked pasta and a splash of reserved pasta water. Sprinkle in the Parmesan cheese and toss until everything is evenly coated and glossy. Adjust seasoning with salt, pepper, and a squeeze of lemon juice if desired.
Serve: Plate the pasta, top with crispy sage leaves, and finish with extra Parmesan cheese.