In a saucepan on medium heat bring the beef broth to a simmer.
In a bowl whisk the flour and water together until the flour dissolves completely and there are no lumps. Pour the mixture into the simmering broth; continue simmering until the broth thickens.
Add the roast beef to the thickened broth and continue simmering for 4 - 5 minutes.
Melt the cheese on the rolls underneath the broiler.
Place the roast beef onto the roll, drizzle with gravy, cut in half, and serve with chips and pickles on the side.