Roast the squash: Heat the oven to 425°F (220°C). Toss the cubed squash with 1 to 2 tablespoons olive oil, salt, and pepper. Spread on a baking sheet and roast for 25 to 30 minutes, flipping once, until tender and caramelized at the edges.

Boil the pasta: Bring a large pot of salted water to a boil. Cook the pasta until just shy of al dente. Reserve at least 1 cup of pasta water, then drain.

Crisp the sage: In a large skillet, heat 1 tablespoon olive oil over medium heat. Add the sage leaves in a single layer and cook 30 to 60 seconds per side until crisp. Transfer to a paper towel and sprinkle with salt.

Build the base: In the same skillet, lower the heat to medium-low and add the butter. Once melted, add the onion with a pinch of salt and cook 5 to 7 minutes until soft and translucent. Stir in the garlic and red pepper flakes; cook 30 seconds until fragrant.

Make the sauce: Add the roasted squash and broth to the skillet. Mash with a spoon or spatula to break it down. Transfer to a blender (or use an immersion blender) and blend until smooth, adding a splash of pasta water to loosen. Aim for a creamy, pourable texture.

Finish the pasta: Return the blended sauce to the skillet over low heat. Add the cooked pasta and toss, splashing in pasta water as needed to coat every piece. Stir in the Parmesan and lemon juice. Taste and adjust with salt and pepper.

Serve: Top with the crispy sage, extra cheese, black pepper, and toasted nuts. Drizzle with a little olive oil if you’d like a glossy finish.