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 2-3 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 twist firmly to squeeze out as much excess moisture as possible. Transfer the dried potato mixture to a mixing bowl and combine with flour, one beaten egg, salt, and black pepper. Mix thoroughly 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 into the skillet, pressing down to flatten into even pancake shapes. Cook for 4-5 minutes per side until deeply golden and crisp. Transfer to a plate lined with paper towels to drain.
04 - In a clean nonstick skillet, melt the butter over medium-low heat. Crack the eggs into the skillet and fry to your desired doneness, sunny side up or over easy works best for this dish. 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, avocado slices, and a scattering of fresh chives.
06 - Serve immediately while hot for the best texture and flavor.