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Remove steak from refrigerator and let come to room temperature, 30 minutes.
Rub steak with 1 tablespoon vegetable oil and season all over with 1 ½ teaspoon salt, 1 teaspoon pepper, and ¾ teaspoon paprika.
Heat a medium cast iron skillet over medium-high heat. Add steak and cook, flipping once, until a deep golden crust begins to form on both sides of the steak, about 5 minutes total.
Reduce heat to medium-low and add butter to pan. Using a kitchen towel, carefully grip the skillet handle and tilt towards you so that the melting butter forms a pool at the bottom of the skillet. Using a spoon, continually baste butter onto steak to form a deeper golden crust. Flip steak and repeat, 2 to 3 minutes in total. Using a meat thermometer, check for doneness: 120-125°F for medium rare, 130°F for medium.
Transfer steak onto a cutting board and let rest about 10 minutes.
Meanwhile, heat remaining oil in a nonstick skillet over medium heat until shimmering. Crack egg into pan and cook 3 to 4 minutes, or until white is set and edges are just beginning to brown. Remove from pan and season with salt and pepper.
Slice steak into 1" strips on a bias, against the grain.
Serve steak with eggs. Garnish with herbs and season with more salt, pepper, and paprika to taste.