Heat the olive oil in a large skillet over medium heat. Add the onions and bell peppers and saute until softened, about 10 minutes. Add the beef, breaking it up with a wooden spoon, and cook until browned, 3 to 5 minutes. Add the tomatoes, beans, tomato paste, vinegar, brown sugar, and some salt and pepper, and cook until slightly thickened, 10 to 15 minutes.
Serve warm on the hamburger buns.
Adapted from "Trisha's Table: My Feel-Good Favorites for a Balanced Life" by Trisha Yearwood © Clarkson Potter 2015. Provided courtesy of Trisha Yearwood. All rights reserved.