Heat a large skillet over medium heat. Add the ground beef and cook for 3–4 minutes, stirring and breaking the meat into small crumbles as some of the fat begins to render.
Add the chopped onion and green bell pepper. Continue cooking for another 3–5 minutes, stirring occasionally, until the vegetables soften and the beef is completely browned and cooked through.
If excess grease has accumulated in the skillet, carefully drain it before adding the sauce ingredients.
Stir in the ketchup, brown sugar, yellow mustard, and garlic powder. Season with salt and freshly ground black pepper to taste.
Reduce the heat to low and allow the mixture to simmer gently for 20–30 minutes, stirring occasionally, until the sauce thickens and coats the ground beef.
Taste the filling and adjust the ketchup, brown sugar, or mustard if needed to achieve your preferred balance of sweet and tangy flavors.
Divide the warm ground beef mixture evenly among the six split hamburger buns. Serve immediately.