Heat oil in a large skillet over medium heat. Once the oil is shimmering, add the sage and toss to coat. Let the sage get crispy before transferring it to a small bowl. Sprinkle it lightly with sea salt and set the bowl aside.

Add squash, onion, garlic and red pepper flakes to skillet. Season with salt and pepper. Cook, stirring sometimes, until onion is translucent, about 8 to 10 minutes. Add broth. Bring the mixture to a boil, then reduce heat and simmer until squash is soft and liquid is reduced by half, about 15 to 20 minutes.

In the meantime, bring a large pot of salted water to a boil and cook the pasta according to package directions, stirring sometimes, until al dente. Drain, reserving 1 cup cooking liquid.

Once the squash mixture is finished cooking, remove it from heat and let it cool barely. Transfer the contents of the pan to a blender. Reserve the skillet. Purée the mixture until smooth (watch out for hot steam escaping from the top of the blender), then season with salt and pepper to taste.

Combine pasta, squash purée, and 1/4 cup cooking liquid in reserved skillet and cook over medium heat, tossing and adding more pasta cooking liquid as needed, until sauce coats pasta, about 2 minutes. Season with salt and pepper if necessary.

Serve pasta topped with fried sage, more black pepper, shaved Parmesan/Pecorino, and smoked salt, if desired.