Heat olive oil in a large skillet over medium-high heat. Add ground beef and cook 6–7 minutes, breaking into small pieces, until well browned. Drain most fat, leaving about 1 tablespoon.
Add onion and bell pepper; cook 4–5 minutes until softened. Add garlic and stir 30 seconds.
Whisk all sauce ingredients together in a small bowl. Pour over the beef mixture and stir to coat. Add broth and stir.
Bring to a gentle simmer. Cook uncovered 10–15 minutes until the sauce thickens and clings to the meat. Taste and adjust seasoning.
Toast buns: Butter cut sides and toast in a dry skillet or under broiler until golden.
Spoon filling generously onto bun bottoms. Add pickles or cheese if using. Serve immediately.