Place a large skillet over medium-high heat. Add the ground beef and break it into small crumbles using a wooden spoon. Cook until browned, about 5-7 minutes. Drain excess fat if necessary, leaving about 1 tablespoon in the skillet.

Add the diced onion and green bell pepper to the skillet. Sauté for 5 minutes until the onions are translucent. Stir in the minced garlic and cook for 1 minute until fragrant.

Stir in the tomato paste, coating the beef and vegetables. Add the ketchup, brown sugar, yellow mustard, and Worcestershire sauce. Season with salt and black pepper.

Add water or beef broth if the mixture is too thick. Reduce heat to low and simmer for 10-15 minutes, stirring occasionally, until the sauce thickens.

While the sauce simmers, toast the hamburger buns with butter in a separate skillet or toaster. Assemble by spooning the sloppy joe mixture onto the toasted buns.