Heat 1 Tbs. of the oil in a large (12-inch) sauté pan over medium heat until it’s shimmering. Add the sausages and cook, turning every couple minutes, until they’re browned all over, about 8 minutes. Transfer to a large plate.
Add the remaining 2 Tbs. oil and the onion to the pot, sprinkle with 1/2 tsp. salt, and cook, stirring occasionally, until the onion softens completely and starts to turn light brown, about 7 minutes. Add the balsamic vinegar and chicken broth, and scrape the bottom of the pot with a wooden spoon to incorporate any browned bits.
Reduce to a gentle simmer (medium-low or low depending on your stovetop). Add the sausages and grapes, cover the pot with the lid ajar, and cook, stirring occasionally, until the sausages are cooked through (slice into one to check), about 25 minutes. Serve immediately, sprinkled with the oregano.