Brown ground beef in skillet until cooked. Transfer beef to a seperate platter. You can leave the grease in the pan if there is a tablespoon or less.
Melt 2 tablespoons of butter in skillet. Add onions and peppers. Cook, stirring occasionally for 3-4 minutes. Onions should look golden, and peppers should be soft.
Pour 1/4 c. beef broth into skillet and stir. This will scrape up any bits of meat and onion from the bottom of the pan.
Add Worcestershire sauce.
Add beef back into pan. Stir.
Add remaining 1/4 c. of beef broth.
Stir in 4 slices of American cheese until melted and evenly distributed. Bring mixture to a simmer and heat until it begins to thicken.
Add cracked pepper and salt, to taste. American cheese is salty, by nature, so add salt sparingly. Remove from heat.
Slice 12 slider rolls through the center (as shown in photo). Grease the bottom of a baking sheet with butter or non stick cooking spray.
Place sliders on baking sheet. Remove slider tops and set to the side.
Spoon filling over roll bottoms. Top with remaining 4 slices of American cheese.
Replace slider tops. Brush with remaining 2 tablespoons of butter (melted).
Bake at 350 degrees for 15 minutes. Let stand 10 minutes before serving.