In a large bowl, combine the beaten egg, condensed French onion soup, dry breadcrumbs, kosher salt, and black pepper with the ground beef. Use your hands or a wooden spoon to mix until everything is fully incorporated. Shape the mixture into six oval-shaped patties.

Heat a large skillet over medium heat. Add the patties and cook until browned on both sides, about 3-4 minutes per side. Note that the patties don’t need to be fully cooked at this stage, as they’ll finish cooking in the gravy.

Remove the patties from the skillet and drain any excess grease. In the same skillet, whisk together the flour, ketchup, water, Worcestershire sauce, and dry French onion soup mix until well combined. Add the condensed cream of mushroom soup and whisk until the mixture is smooth and creamy.

Return the patties to the skillet and spoon the gravy over them. Cover the skillet and let it simmer over low heat for 20-25 minutes, or until the patties are cooked through and the gravy is thick and bubbly.