In a large, deep skillet, heat 2 tbsp. olive oil. Add squash, season lightly with salt and cook over high heat, stirring until tender and lightly browned, about 8 minutes. With slotted spoon, transfer to a plate. Add remaining 1 tbsp. of oil to skillet. Add onion and sausage and cook over high heat, breaking up sausage with a spoon until lightly browned and cook thru. Returned squash to skillet.
Cook pasta and drain, reserving 1 cup of pasta water. Add pasta to skillet along with reserved water and the grated cheese. Cook over high heat, stirring until sauce is thick and creamy, 1-2 minutes. Stir in fresh basil.