In a large bowl, combine the beaten egg, 1/3 cup of condensed French onion soup, dry breadcrumbs, kosher salt, and black pepper to the ground beef. Mix well with your hands or a large wooden spoon until fully incorporated.

Shape the beef mixture into six oval-shaped patties.

In a large skillet, cook the patties over medium heat until they are browned on both sides, about 3-4 minutes per side. The patties will not be fully cooked, but that's okay as they will cook further later on.

Remove the patties from the skillet and set aside. Drain off any excess grease from the skillet.

In the same skillet, whisk together the all-purpose flour, ketchup, water, Worcestershire sauce, and packet of dry French onion soup mix until well combined.

Add the can of condensed cream of mushroom soup to the skillet and whisk until the mixture is smooth and well combined.

Return the patties to the skillet and spoon the gravy over them.

Cover the skillet and simmer over low heat for 20-25 minutes or until the patties are cooked through and the gravy is thick and bubbly.