1Place the beef in a large, deep skillet over medium heat, and cook until evenly brown. Transfer beef to a strainer over a saucepan, and allow grease to drain for about 5 minutes. Mix any drippings remaining in skillet into the saucepan. There should be about 1/2 cup drippings. Return beef to skillet.
2Heat the beef drippings in the saucepan over medium heat, and gradually mix in 1/4 cup flour. Reduce heat to low, and continue cooking 10 minutes, stirring continuously, to form a golden brown roux. Pour in the beef broth, and remove from heat.
3Pour the water into the skillet with the beef, and mix in remaining flour. Stir in the roux mixture, chili powder, carrot, vinegar, onion, salt, sugar, paprika, and garlic powder. Bring to a boil, reduce heat to medium-low, and continue cooking 15 minutes, until thickened. When it's done cooking, take the chili off the heat, cover it, and let it sit for 30 minutes before using it on burgers, etc. It should thicken to a tasty brown paste as it sits.