Heat a large skillet over medium heat. Add the ground beef and chopped onion. Cook until the beef is browned and the onion is soft, breaking up the meat as it cooks.
Stir in the tomato sauce, brown sugar, and Worcestershire sauce. Season with salt and pepper. Simmer for 5 to 7 minutes until the mixture thickens slightly.
While the meat simmers, mix the shredded cabbage, grated carrot, mayonnaise, lemon juice, and sugar in a bowl. Add salt and pepper to taste. Set the coleslaw aside.
Spread butter on the cut sides of the hamburger buns. Place them butter-side down in a separate skillet or on a griddle over medium heat. Toast until golden, about 2 minutes.
Spoon the sloppy joe mixture onto the bottom halves of the toasted buns. Top each with a generous scoop of creamy coleslaw and add the bun tops.