Begin making the Carolina Chili. (To avoid greasy chili, be sure to choose a very lean ground beef since the beef is not browned and drained in this recipe). Place the beef and 2 cups of water in a 4 1/2-quart Dutch oven or soup pot over high heat. Bring the water to a boil.
Add the chopped onion to the beef. When the water boils again, reduce the heat to medium. Stir to begin breaking up the meat.
Add the tomato paste, ketchup, chili powder, Worcestershire sauce, vinegar, salt, and pepper, if using. Stir well until the tomato paste has dissolved and the meat is mostly broken up.
Continue to cook the chili at a slow boil, stirring about every 5 minutes or so, until the mixture is thick, about 15 minutes. As the mixture thickens, you may need to reduce the heat to medium-low or low to prevent sticking.
Many Southerners prefer chili that is finely ground (roughly the consistency of cooked oatmeal) because it's easier to spread on a hot dog and easier to eat as well. To try it this way, place the chili (cooled at least 20 minutes) in a blender or food processor and blend on high until finely ground, 30 to 45 seconds.
While the chili simmers and about 12 to 15 minutes before you plan to serve the chili dogs, turn on a gas grill to medium-high.
Place the hot dogs on the hot grill and cook until heated through and just beginning to brown all over, about 5 minutes. Heat the buns on the grill, if desired. To serve, place each hot dog in a bun and smother with chili.
Make-ahead: The chili can be refrigerated, covered, for up to 2 days or it can be frozen for up to 2 months. * If the chili has been frozen, thaw it in a microwave oven on the defrost setting, following the manufacturer's directions and stirring after each 1-minute interval, then reheat. * To reheat, place the chili in a microwave-safe container, cover with a paper towel, and microwave on high until heated through, 2 to 3 minutes for 1 cup, stopping once halfway through.