Preheat the oven broiler on low.
Don't separate the buns individually. Slice the buns in half creating a top sheet and a bottom sheet. Melt half of the butter in a nonstick pan on medium heat, take the top sheet of the buns and place into the pan, cook until golden brown then remove and set aside. Do the same with the bottoms of the buns using the remaining butter. Then remove and set aside.
In a saucepan on medium heat bring the broth to a simmer. Add in the turkey and cover with a lid. Low simmer until the broth concentrates in flavor. Remove the turkey and place on the bottom bun sheet and top with cheese. Season the broth with salt and pepper to taste.
In a bowl whisk the flour with an equal amount of water (or more if needed) until the flour dissolves completely and there are no lumps. Slowly pour the mixture a little at a time into the simmering broth; continue simmering until the broth thickens and the flavor concentrates.
Place the turkey under the broiler to melt the cheese. Once melted place the top sheet on and cut into portions.
Serve with a side of gravy, chips and pickles.