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 just smoking. Sear the steak for 2 to 3 minutes per side for medium-rare, adjusting time for your preferred doneness. Transfer to a cutting board and rest for at least 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 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 tablespoons 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 about half an inch thick. Cook for 4 to 5 minutes per side until deep golden brown and crisp throughout. Drain briefly on paper towels.
04 - Melt the butter in a nonstick skillet over medium-low heat. Crack the eggs into the pan and fry to your preferred doneness, sunny side up or over easy works best for the stacks. Season lightly with salt and pepper.
05 - Place one hash brown on each serving plate. Layer with sliced steak and a generous sprinkle of cheddar cheese. Top with a fried egg, then finish with sour cream, green onions, avocado slices, and fresh chives as desired.
06 - Serve immediately while hot so the cheese melts slightly and the hash browns remain crisp.