In a large bowl, combine all ingredients for the beef patties except the oil. Mix gently until everything is just incorporated, then shape into evenly sized patties, slightly flattening the centers.
Heat oil in a skillet over medium heat. Add the patties and cook until both sides are golden brown and fully cooked through. Remove from the skillet and keep them covered to stay warm.
In the same skillet, sauté the diced onion until it starts to soften. Add garlic and butter, stirring until fragrant and melted.
Add the sliced mushrooms and cook until they are browned and tender. Season lightly with salt and pepper to enhance flavor.
In a separate bowl, whisk together the flour, beef broth, water, Worcestershire sauce, Dijon mustard, ketchup, salt, and pepper until smooth. Pour this mixture into the skillet with the mushrooms.
Let the gravy cook until it thickens, stirring occasionally to keep it smooth and lump-free.
Return the cooked patties to the skillet and spoon the gravy over them. Let everything cook together briefly until the patties are heated through and coated in the rich sauce.