Preheat a cast-iron skillet over medium-high heat for 5 minutes.
Add 1 tablespoon avocado or vegetable oil to skillet and heat until shimmering.
Cut steak into two even pieces; pat dry with paper towels then season with 1 teaspoon salt.
Add steak to skillet and cook until deeply browned on both sides and internal temperature reaches 125ºF, about 3 minutes per side for medium rare. (See recipe notes for different temperature doneness.)
Transfer steak to a plate and brush with some of gochujang sauce; let rest 5 minutes.
Meanwhile, add 1 tablespoon butter to cast-iron skillet (with steak drippings) set over medium heat.
Once butter has melted and is starting to bubble, carefully crack 4 eggs into skillet. Cook eggs, carefully tilting skillet so oil pools. Use a large spoon to scoop oil up and spoon over egg whites. Continue basting until whites are completely set and yolk is cooked to desired doneness, 2–3 minutes. (If you like a runny yolk, avoid spooning oil over yolks. If you like a firmer yolk, spoon oil over entire eggs.)
Slice steak against the grain and transfer to plates; brush additional sauce over steak slices. Add eggs and pickled vegetables to plate; sprinkle scallion greens over top and serve.