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 5 minutes before slicing thinly against the grain.
02 - Place the grated potatoes and onion in the center of a clean kitchen towel and wring out as much moisture as possible. Transfer the squeezed mixture to a mixing bowl and add the flour, egg, salt, and pepper. Stir until everything is evenly combined and the mixture holds together.
03 - Heat 2 to 3 tbsp vegetable oil in a large nonstick skillet over medium-high heat. Scoop roughly one quarter of the potato mixture for each hash brown and press into flat pancake shapes in the pan. Cook for 4 to 5 minutes per side until deeply golden and crispy. 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 here. Season with a pinch of salt and pepper.
05 - Set one crispy hash brown on each plate. Layer on the sliced steak, a generous sprinkle of cheddar cheese, and a fried egg. Finish with sour cream, green onions, avocado slices, and a scattering of fresh chives.
06 - Serve the stacks immediately while everything is still hot and the cheese is beginning to melt from the warmth of the steak and eggs.