In large skillet, over medium-high heat, add the ground beef and brown for about 5 minutes. Add the onion and green pepper and cook another 3-4 minutes, until the vegetables start to soften. Add the steak sauce and beef broth, season with salt and pepper, bring to a simmer and cook 4 minutes.
Prepare the buns as the meat simmer. Toast in the oven or flat top. The buns will brown quickly so keep a close eye on the buns.
Turn meat temperature to low and set the toasted buns aside. Melt the butter in a medium-size pot over medium-high heat. Stir in the flour and cook about 30 seconds. Whisk in milk, and simmer to let it thicken for about 2 minutes. Turn off heat and stir in the cheese.
To serve, place a scoopful of the meat mixture onto the bottom of the buns and top with the cheese sauce. Serve.