Grate onion on the large holes of a grater. Mince the garlic. Finely chop (or grate) the carrot. Dice bell pepper. OR: Add the carrot, bell pepper, and garlic to a food processor and pulse until finely chopped.
Heat oil and butter in a large pan over medium-high heat. Add onion, garlic, carrot, and bell pepper. Cook for 6–8 minutes, or until most of the liquid has evaporated and veggies are mostly tender. Push veggies to the edges. Add ground beef to the center. Let sit for 45 seconds, then stir and break it up. Cook for 5–8 minutes, or until no longer pink. Mix the veggies into the beef as it cooks.
Stir in the tomato paste and crushed tomatoes. Cook for 2–3 minutes to deepen the flavor. Add ketchup, Worcestershire, brown sugar, ground mustard, beef bouillon, and pepper. Stir well, then reduce heat to low. Simmer uncovered for 15 minutes, stirring often, until the sauce thickens.
While the sauce simmers, prepare the buns (see note 2). Once sauce has thickened, stir in red wine vinegar if using. Taste and adjust seasoning. Remove the pan from heat. Spoon the filling onto toasted buns and serve right away.
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Notes
Note 1: Depending on the tomatoes and ketchup (some are sweeter than others), you may want more or less brown sugar.Note 2: To toast the buns, spread softened butter on the inside and place them in a dry nonstick pan over medium-high heat for 1–3 minutes, until golden.Storage: Store the meat in a sealed container in the fridge for 3–5 days or freeze for up to 3 months. Reheat on the stove or in the microwave until hot. Keep buns separate and toast just before serving.