Cook and stir half of the shaved steak in a large skillet over medium heat until browned, 3 to 5 minutes. Transfer to a plate. Repeat with remaining steak. Drain juices from the steak; cut into small pieces.

Combine steak, American cheese, green bell pepper, red bell pepper, and yellow bell pepper in a bowl. Season with salt and pepper.

Lay 1 egg roll wrapper on a flat surface in the shape of a diamond. Place a large spoonful of the steak mixture in the center. Fold the bottom corner and opposing sides over the steak mixture; roll up wrapper towards the top corner. Repeat with remaining egg roll wrappers and steak mixture.