Heat olive oil in a large, deep skillet over medium heat. Cook and stir onion with a pinch of salt in hot oil until onion is soft and golden, 5 to 7 minutes. 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 mixture and bring to a boil while scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Add orecchiette pasta; cook and stir pasta in hot broth, adding remaining broth when liquid is absorbed, until pasta is cooked through and most of the broth is absorbed, about 15 minutes.
Stir arugula into pasta-sausage mixture until arugula wilts. Ladle pasta into bowls and dust with Parmigiano-Reggiano cheese.