Heat oil in the pot over medium heat. Shape sausage into 3/4-inch balls and add them to the pot. Cook, turning occasionally, until browned all over, about 5 minutes. Add onion and garlic and cook, stirring frequently, until the onion is translucent, 5 to 8 minutes. Stir in broth and the reserved cooking liquid, scraping up any browned bits. Increase heat to high and bring to a boil.