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 shimmering. Sear the steak for 2-3 minutes per side for medium-rare, adjusting time to your preferred doneness. Transfer to a cutting board and let 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, then twist and squeeze firmly to extract as much moisture as possible. Transfer the dried potato mixture to a mixing bowl and combine with flour, one beaten egg, salt, and black pepper. Mix until evenly incorporated.
03 - Heat 2-3 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 in the skillet, pressing down gently to flatten into even pancake shapes. Cook for 4-5 minutes per side until deeply golden and crispy. Drain on paper towels and repeat with remaining mixture.
04 - Wipe the skillet clean and melt butter over medium-low heat. Crack the eggs into the pan and fry to your preferred doneness — sunny side up or over easy work best for these stacks. Season lightly with salt and pepper.
05 - Place one crispy 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 the stacks immediately while hot for the best texture and flavor.