Heat olive oil in a large, deep skillet over medium heat. Cook and stir onion in hot oil until onion is soft and golden, 5 to 7 minutes. Season with salt to taste. Stir sausage into onions; cook and stir until sausage is broken up and browned, 5 to 7 minutes.
Pour 1 1/2 cups chicken broth into sausage and bring to a boil while scraping up the brown bits. Add pasta. Cook and stir in hot broth, adding remaining broth when liquid is absorbed, continue stirring until pasta is cooked through and most of the broth is absorbed, about 15 minutes.
Stir spinach into pasta-sausage mixture until it wilts. Ladle pasta into bowls and top with Parmigiano-Reggiano cheese.