In a large bowl, mix together the grated onion (and its juices), the breadcrumbs (1/3 cup), the egg, the kosher salt (1 teaspoon), and the black pepper (1/2 teaspoon) until well combined.
Add the ground beef to the bowl and mix to combine with the breadcrumb and onion mixture. I recommend using your hands for this.
Divide the beef mixture evenly into four. The easiest way to do this is to press the beef mixture into the bottom of the bowl and use your finger to “draw” lines dividing it into four quadrants. Form 4 oval shaped patties with your hands about 3/4ths of an inch thick.
Heat a large skillet over high heat. Only after it’s searing hot (after about 2 minutes), add the olive oil (1 tablespoon) to the pan and swirl to coat.
Add the patties to the hot pan and cook for 2 minutes on each side- they should a have nice brown color. To get the best color, don’t move them until the two minutes are up.
Remove the patties to a plate.
Add the butter (2 tablespoons) to the skillet, then add the sliced mushrooms after it’s melted.
Cook the mushrooms for 2 minutes, until golden. Then, sprinkle the flour (2 tablespoons) over the mushrooms. Stir the mushrooms, coating each of them in the flour, and continue to cook for 1 more minute.
Add the beef broth (2 cups), stirring constantly to incorporate the flour, and bring it to a simmer.
Add the steak patties back into the gravy mixture and simmer, uncovered, for 5 more minutes, or until they’re fully cooked and the gravy has thickened.
