Bring a large pot of salted water to a boil. Cook orecchiette according to package instructions. Drain, reserving 3/4 cup starchy cooking water, and set pasta aside until ready to use. Meanwhile, heat 1 tablespoon olive oil in a large skillet on medium-high heat. Add sausage, crumbling with the back of a spatula, and cook until browned. Carefully remove cooked sausage from skillet and transfer to a paper towel lined plate. Heat remaining olive oil in the same skillet on medium heat. Add onion and sauté until translucent, about 3 minutes. Stir in garlic cloves and cook for a minute. Add broth to the pan, scraping up any browned bits at the bottom to deglaze. Cook until liquid is mostly absorbed. Add mustard greens to the skillet along with 1/4 cup of water. Stir, cover and cook 6 minutes until greens are tender and wilted. Uncover, add sausage, and continue to cook until most of the water is absorbed. Stir in cooked pasta and enough starchy cooking liquid to make a sauce. Serve garnished with grated cheese.
