Heat olive 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 the squash, onion, garlic and red pepper flakes to the skillet. Season with salt and pepper. Cook, stirring occasionally, until the onion is translucent, about 8 to 10 minutes. Add the broth. Bring to a boil, then reduce the heat and simmer until the squash is soft and the liquid is reduced by half, about 15 to 20 minutes.
Meanwhile, bring a large pot of salted water to a boil and cook the pasta, stirring occasionally, until al dente according to package directions. Reserve 1 cup of the pasta cooking water before draining.
When squash is done, remove it from the heat and let it cool slightly. Purée in a blender or using an immersion until smooth. Season with salt and pepper to taste. Place back in the skillet, along with the pasta and ¼ cup cooking liquid. Cook over medium heat, tossing and adding more pasta cooking water as needed, until the sauce coats the pasta, about 2 minutes. Season with more salt and pepper, if necessary.
Divide between two plates and top with crispy sage, shaved Parmesan and more salt, if desired.