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 or heavy-bottomed skillet over high heat until the oil just begins to smoke. 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 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 twist firmly to squeeze out as much excess moisture as possible. Transfer the dried potato and onion mixture to a mixing bowl. Add the flour, one beaten egg, salt, and black pepper. Stir until everything is evenly combined and the mixture holds together.
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 place it in the skillet, flattening it into a round pancake shape about 1/2 inch thick. Cook for 4 to 5 minutes per side until deeply golden and crisp. Transfer to a plate lined with paper towels to drain.
04 - Wipe out the skillet or use a separate nonstick pan. Melt the butter 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 with a pinch of salt and pepper.
05 - Place one crispy hash brown on each serving plate. Layer sliced steak over the hash brown, then sprinkle with shredded cheddar cheese so it begins to melt from the residual heat. Crown each stack with a fried egg, then finish with sour cream, sliced avocado, green onions, and fresh chives as desired.
06 - Serve the stacks immediately while everything is still hot and the hash browns remain crispy. Pair with bold coffee or fresh orange juice for a complete brunch experience.