Heat a large cast-iron skillet over medium-high heat. Generously sprinkle the filets all over with salt and pepper, including on the sides. Add the oil to the pan. When the oil is nearly smoking, add the filets and cook for 1 minute. Add a third of the butter and cook for about 3 minutes. Flip the filets and cook another 3 minutes or so for medium-rare, occasionally spooning melted butter over the top. Using tongs, brown each filet on its side, then transfer to a cutting board to rest.
Meanwhile, heat a medium nonstick skillet over medium heat. Immediately add the remaining butter and the eggs; cook undisturbed just until the eggs begin to set at the edges, about 40 seconds. Then begin to stir the eggs gently to form large curds. Continue cooking until the eggs are just barely set, with some large curds surrounded by a bit of liquid egg, 45 to 60 seconds. Season with salt. Drop pieces of goat cheese and spoonfuls of romesco sauce over the top and sprinkle with parsley. Stir quickly just to marble in the romesco; the eggs should still be very softly set. Remove from the heat.
Working quickly, remove the strings and slice each filet into 3 pieces; arrange on 4 plates. Spoon some eggs next to the meat, and top with a drizzle of olive oil and some more parsley. Serve immediately.