In a large skillet, brown the ground beef over medium-high heat until it is browned and cooked through. Remove the ground beef from the heat and drain off the fat (retaining about a tablespoon to fry the vegetables). Set the ground beef aside.
Add the diced onions and green peppers to the same skillet and saute until tender, 10 to 15 minutes.
While the vegetables are cooking, whisk together the tomato paste, tomato sauce, barbeque sauce, brown sugar, ground mustard, vinegar, and salt, in a small bowl. Set aside.
Add the minced garlic to the skillet and saute with the vegetables for an additional 30 seconds, stirring constantly.
Return the cooked ground beef to the skillet and stir together with the vegetables. Turn the heat down to medium and add the sauce ingredients. Stir until completely mixed in. Taste and add more salt if needed.
Continue cooking until everything is heated through and steaming, about 10 minutes.
Serve on toasted buns (see notes).