Remove all the leaves from the rosemary sprigs except the top inch; reserve the leaves for another use.
Skewer the tomatoes on 3 rosemary sprigs. Repeat with the onions, garlic, and beef, dividing the ingredients evenly among the remaining sprigs and using separate sprigs for each ingredient.
Drizzle each skewer with oil and season with the salt and pepper. Transfer the garlic and onion skewers to the grill. Grill, turning occasionally, until softened, about 15 minutes. After about 8 minutes, add the tomato and beef skewers to the grill. Cook the beef, turning once, until the desired doneness, 7 minutes for medium-rare. Cook the tomatoes, turning once, until the skins burst, about 7 minutes. Transfer the skewers to a baking sheet when they are done.
Remove and discard the papery skins from the garlic. Using a fork, slide the garlic and tomatoes into a small bowl. Add the vinegar and combine, mashing the tomato and garlic with the fork until chunky. Serve the tomato chutney alongside the beef and onion skewers.
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