1. In a large bowl, combine the egg, oats, 2 tablespoons onion, ¼ teaspoon salt and ⅛ teaspoon pepper and mix well. Crumble the ground beef over the egg mixture. Using your hands, mix well. Form into 2 patties about 3 inches wide and ¾ inch thick.
2. Heat 1 tablespoon oil in a regular (not nonstick) skillet over medium-high. Add patties and cook about 3 minutes on each side or until golden brown. Remove to a plate.
3. In the same skillet, add the remaining onion and mushrooms. Cook until onions are tender and mushrooms have released their water, about 3 minutes. Scrape onto plate with beef patties.
4. Reduce heat to medium-low and add 1 to 3 teaspoons oil, enough to cover the bottom of the skillet in a thin pool. Sprinkle the flour over the oil and stir well, forming a thick paste. Continue to cook the paste until it turns a medium brown and all the bits and pieces of the beef are incorporated in the paste. Slowly and carefully add the water, stirring constantly to form the gravy.
5. Once the gravy comes to a low boil and begins to thicken, add the beef patties and the vegetables back into the skillet. Turn the heat to low and spoon the gravy over the meat. Simmer 5 to 15 minutes, until ready to serve.
6. Add a few more tablespoons water if gravy becomes too thick. Taste and adjust seasoning with additional salt and pepper, if desired. Serve.
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