Heat a skillet over medium-high heat.

Add ground beef, chopped onion, chopped green bell pepper, finely chopped carrots, and minced garlic to the skillet.

Cook and stir until the beef is browned, about 10 minutes, and the vegetables are softened.

Drain off any excess liquids.

Stir in prepared yellow mustard, ketchup, and brown sugar.

Mix thoroughly to combine all ingredients.

Reduce heat to low.

Simmer the mixture until the sauce has thickened, about 30 minutes.

Season with salt and pepper to taste.

Serve hot on buns.