Add the oil to a large skillet over medium-high heat. Add the ground beef and cook 5-6 minutes or until no longer pink. Add the onion and bell pepper and continue to cook 2-3 minutes or until crisp tender. Drain off any fat. Stir in the steak sauce and beef broth (and Worcestershire, if using). Season with salt and pepper. Cook until liquid is gone.
Butter the cut sides of the buns. Toast the buttered sides in a hot skillet on medium-high heat. Set aside.
To make the cheese sauce, melt the butter in a 1-quart saucepan over medium-high heat. Whisk in the flour and cook about 1 minute. Whisk in milk, bring up to a bubble and cook until thickened. Remove from heat and stir in the cheese. Season with salt and pepper, if desired.
To serve, place a scoop of the meat mixture onto the bottom half of the buns. Top with cheese sauce (or cheese) and cover with the top half of the bun.