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 for 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. Gather the corners and twist firmly to squeeze out as much excess moisture as possible. Transfer the squeezed potato and onion mixture to a mixing bowl and add the flour, egg, salt, and black pepper. Stir until evenly combined.
03 - Heat 2 to 3 tbsp 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, flattening each portion into a round pancake shape about 1/2 inch thick. Cook for 4 to 5 minutes per side until deeply golden and crisp. Drain on paper towels.
04 - Melt the butter in a nonstick skillet 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 the stacks. Season lightly with salt and pepper.
05 - Place one hash brown on each serving plate. Layer with sliced steak, a generous sprinkle of cheddar cheese, and a fried egg. Finish with sour cream, green onions, avocado slices, and fresh chives as desired.
06 - Serve immediately while hot for the best texture and flavor.