Preheat your oven to 325°F. Warm up the oil in a small saucepan over medium heat. Add the finely chopped onion and cook, stirring, until it becomes tender, about 4-6 minutes. Incorporate the minced garlic and cook for an additional minute. Mix in the ketchup, tomato paste, brown sugar, water, vinegar, mustard, salt, and pepper. Bring everything to a boil, then reduce the heat and let it simmer uncovered until the sauce thickens slightly.

Reserve 1/2 cup of the sauce and refrigerate it. Place the beef short ribs in a 13x9-inch baking dish. Brush the remaining sauce over both sides of the ribs. Cover the dish tightly with foil and bake until the ribs are tender, which should take about 2 to 2 1/2 hours.

Set your broiler to high. Remove the juices from the baking dish and brush the reserved 1/2 cup of barbecue sauce over the ribs. Broil the ribs uncovered until the sauce becomes bubbly and slightly caramelized, which should take about 2-3 minutes.