In a large bowl combine the cabbage and carrot mix and the 2 tablespoons of minced onion. In a separate bowl combine the sugar, ½ teaspoon of salt, 1/8 teaspoon pepper, milk, mayonnaise, buttermilk, 1 ½ tablespoon white wine vinegar, and lemon juice, mix until smooth. Pour the mixture over the coleslaw and onion and stir well. Cover and chill in the refrigerator for 1 hour.
Place the beef and the water into a soup pot over high heat. Bring the water to a boil. Add the chopped onion. Reduce heat to medium.
Stir to begin to break up the meat. Add the tomato paste, ketchup, chili powder, Worcestershire sauce, 1 teaspoon vinegar, salt, and pepper. Stir until well combined. Continue to cook the chili at a low boil, stirring about every 5 minutes, until the mixture has thickened (about 15 minutes).
Place the hot dogs onto a hot grill over medium-high heat and cook until heated through and beginning to brown all over about 5 minutes. Place each hot dog in a bun, top with some of the chili, coleslaw, and mustard.