In large skillet over medium-high heat, heat oil and then brown the ground beef until completely cooked through and juices run clear.
Add the onion and green pepper to the beef and cook another 3-4 minutes, until the vegetables start to become tender.
Stir in the steak sauce and beef stock. Season with salt and pepper. Bring to a simmer and cook about 2 minutes.
Split open rolls and remove some of the soft insides, leaving a bed to hold the meat mixture. Toast the rolls and set aside.
While the meat is cooking, melt the butter in a medium sauce pan. When butter is melted, stir in the flour and cook about 1 minute until a roux is formed. Whisk in milk and let sauce thicken to desired consistency, stirring constantly. Turn off heat and stir the cheese.
To serve, place a scoopful of the meat mixture onto the bottom of the rolls, then top with the cheese sauce and replace the top of the roll.