In a large bowl, combine egg, ⅓ cup of soup, breadcrumbs, salt, and pepper. Add beef and using a large spoon or your hands, mix to combine. Do not over-mix.
Divide meat mixture evenly to create six patties.
In a large skillet on medium heat, brown patties for 3-4 minutes per side (they will not be fully cooked). Remove patties from the skillet and set aside.
Drain excess grease. To the same skillet on medium heat, add remaining soup, French onion soup mix, ketchup, and Worcestershire sauce, and mix to combine.
In a small bowl, combine flour with water to create a slurry. Pour slurry into the skillet and stir. Cook 2-3 minutes to thicken. Once the sauce thickens, remove about 1 cup of the gravy and set aside.
Add meat patties to the skillet and top with reserved gravy. Cover and cook on low for 15-20 minutes, or until beef patties are fully cooked. Check occasionally to make sure the patties are not sticking. Add ¼ cup water if needed.
Notes
Leftovers will keep in an airtight container in the refrigerator for 4 days and in the freezer for 1 month.