01 - Pat the steak dry with paper towels and season both sides generously with salt and freshly ground black pepper. Heat olive oil in a cast iron skillet over high heat until shimmering. Sear the steak for 2 to 3 minutes per side for medium-rare, adjusting time to your preferred doneness. Transfer to a cutting board and rest for 5 minutes before slicing thinly against the grain.
02 - Place the grated potatoes and onion in the center of a clean kitchen towel. Gather the edges and squeeze firmly to extract as much excess moisture as possible. Transfer the dried potato and onion mixture to a mixing bowl. Add the flour, egg, salt, and black pepper, then stir until evenly combined.
03 - Heat 2 to 3 tbsp of vegetable oil in a large nonstick skillet over medium-high heat. Scoop approximately one quarter of the potato mixture for each hash brown and press into flat pancake shapes in the pan. Cook for 4 to 5 minutes per side until deeply golden and crisp. Drain on paper towels and repeat with remaining mixture.
04 - Melt the butter in a nonstick skillet over medium-low heat. Crack the eggs into the pan and fry to your desired doneness, sunny side up or over easy recommended for the best presentation. Season lightly with salt and pepper.
05 - Place one crispy hash brown on each serving plate. Layer with sliced steak, a generous sprinkle of cheddar cheese, and one fried egg. Finish with sour cream, green onions, sliced avocado, and fresh chives as desired.
06 - Serve the stacks immediately while hot for the best texture and flavor.