Heat BBQ up to a medium heat, alternatively, use a griddle pan. Grill the sausages, turning every couple minutes, until they’re browned and cooked through, about 15 minutes. Transfer to a large plate.
Add the olive oil and onion to a pot and place over medium heat, stirring occasionally. Once the onions have softened and started to lightly brown, add the balsamic vinegar and chicken broth.
Reduce to a gentle simmer and add the grapes until the most of the liquid has evaporated and you have a glossy sauce, about 15 minutes.
Serve the sausages with the sauce spooned alongside a piece of crusty bread and a green salad. Finish with a sprinkle of the oregano.