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 while peeling and coarsely chopping the onion. Add the chopped onion to the beef. When the water boils, 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. Serve on hot dogs (see notes).
NOTE: To avoid greasy chili, choose a very lean ground beef since the beef is not browned and drained in this recipe.
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.
Do 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 to stir.