To start with, set your oven to broil while you are making the sloppy joes. Lay out a 9 x 13 pan and split the potato rolls open. Line the pan with the bottoms of all of the potato rolls and set aside.
Add ground beef to a large skillet and cook over medium-high heat until meat has browned. Drain grease and set meat aside. Using the same skillet, add butter and then sliced onions and green peppers. Cook over medium heat, stirring often, until onions are translucent and peppers are browned a bit.
Add meat back into skillet and add in ketchup, Worcestershire, salt and pepper, apple cider vinegar, dijon, and spices. Mix together cornstarch and beef stock and pour into skillet, stirring to combine.
Next, stir in the Parmesan cheese and then spoon Philly Cheesesteak Sloppy Joe mixture evenly over the pan of roll bottoms. Cover with sliced American cheese and cover with the tops of the potato rolls.
In a microwave safe bowl, melt 1/4 cup butter. Brush tops of rolls with butter and sprinkle 1 -2 tablespoons of Parmesan on them. Broil for 1-3 minutes under oven broiler. Once browned, remove the Philly Cheesesteak Sloppy Joe Sliders from oven and enjoy!