Peel and cube the russet potato, then place in a pot of salted water and bring to a boil; cook until tender, about 15 minutes.
While potatoes cook, combine ground beef, egg, quick oats, garlic powder, sea salt, and black pepper in a bowl, mixing gently until just combined to ensure tenderness.
Form the beef mixture into two large oval patties.
Heat ghee in a large skillet over medium-high heat and sear the patties for 4-5 minutes per side until browned; remove patties from the pan and set aside.
In the same skillet, add sliced onions and mushrooms, sautéing for 5 minutes until softened and caramelized.
Whisk the arrowroot powder into the cold beef bone broth until dissolved, then pour the mixture into the skillet with the vegetables.
Stir the gravy constantly until it reaches a medium-thick consistency, then return the beef patties to the pan to coat in the sauce.
Steam the green beans for 5-6 minutes until bright green and tender-crisp.
Drain the potatoes and mash them with the grass-fed butter and whole milk until smooth.
Serve the hamburger steaks topped with plenty of mushroom gravy alongside the mashed potatoes and green beans.