Brown the beef. Heat a large skillet over medium-high heat. Add the ground beef and cook, breaking it into small crumbles, until no pink remains, about 6–7 minutes. Drain off most of the fat, leaving about a tablespoon in the pan.
Soften the vegetables. Add the diced onion and bell pepper to the beef. Cook 5 minutes, stirring occasionally, until softened. Stir in the garlic and cook 30 seconds more until fragrant.
Build the sauce. Stir in the tomato paste and cook 1–2 minutes until it darkens slightly — this deepens the flavor. Add the ketchup, Worcestershire sauce, brown sugar, mustard, chili powder, salt, pepper, and water. Stir well to combine.
Simmer. Bring to a gentle simmer, then reduce the heat to medium-low. Cook 10–12 minutes, stirring occasionally, until the sauce is thick and glossy. It should coat the meat, not pool around it. Taste and adjust the sugar or salt.
Toast the buns. While the sauce simmers, spread a little butter on the cut sides of the buns and toast them in a dry skillet or under the broiler until golden.
Serve. Pile a generous scoop of the meat mixture onto each toasted bun and serve hot.