Heat oil in a large skillet over medium-high heat. Add pancetta, reduce heat to medium, and cook, stirring occasionally, until crisp, 8–10 minutes. Add sage and toss to coat. Using a slotted spoon, transfer pancetta and sage to a small bowl; set aside.

Add squash, onion, and garlic to skillet; season with salt and pepper and cook, stirring occasionally, until onion is translucent, 8–10 minutes. Add broth. Bring to a boil, reduce heat, and simmer until squash is soft and liquid is reduced by half, 15–20 minutes. Let cool slightly, then purée in a blender until smooth; season with salt and pepper. Reserve skillet.

Cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain, reserving 1 cup pasta cooking liquid.

Combine pasta, squash purée, and ? cup pasta 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. Mix in ? cup Pecorino; season with salt and pepper.

Serve pasta topped with reserved pancetta and sage, shaved Pecorino, and more pepper.

Do Ahead: Squash purée can be made 3 days ahead. Let cool; cover and chill.