Brown the ground beef in a large skillet over medium-high heat, breaking it up as it cooks, about 6 minutes. Drain off the fat but leave about 1 tablespoon in the pan.
Add the onion and bell pepper and cook for 5 minutes, until softened. Stir in the garlic and cook 30 seconds more.
Push the meat to one side and add the tomato paste to the bare part of the pan. Cook it for 1 minute, stirring, until it darkens to a brick color, then mix it through the beef.
Add the ketchup, beef broth, brown sugar, Worcestershire, mustard, chili powder, and salt. Stir to combine and bring to a simmer.
Simmer uncovered for 10 to 12 minutes, stirring now and then, until the sauce thickens enough to mound on a spoon.
Take the skillet off the heat and stir in the apple cider vinegar. Taste and adjust salt.
Butter and toast the buns in a dry skillet, then spoon the filling generously onto each one.