Combine rice and its 1 1/2 cups water in a saucepan. Bring to a boil, cover and cook over low heat for 15 to 18 minutes; rest covered for 5 minutes.
Meanwhile, brown beef and onion in a skillet over medium-high heat for 7 to 8 minutes until the meat reaches 160°F. Drain excess fat.
Stir in ketchup, tomato paste, Worcestershire, filling vinegar, brown sugar, filling water, 1/4 teaspoon salt and pepper. Simmer for 10 to 12 minutes until thick and spoonable.
Whisk mayonnaise, slaw vinegar, granulated sugar and remaining salt in a bowl. Toss with cabbage and carrot just before serving.
Divide rice among four bowls, spoon the hot sloppy joe filling over it and add a small portion of slaw alongside the beef in each bowl.