Prepare the grill for direct cooking over high heat (450° to 550°F).
Lightly brush the steaks on both sides with oil and season evenly with salt, pepper, and the paprika. Allow the steaks to stand at room temperature for 15 to 30 minutes before grilling.
Brush the cooking grates clean. Grill the steaks over direct high heat, with the lid closed as much as possible, until cooked to your desired doneness, 6 to 8 minutes for medium rare, turning once or twice (if flare-ups occur, move the steaks temporarily over indirect heat). When you turn the steaks, place a 12-inch cast-iron skillet on the cooking grate to preheat. Remove the steaks from the grill and let rest for 3 to 5 minutes. Lower the temperature of the grill to medium heat (350° to 450°F).
Coat the skillet with the butter. Crack the eggs into the skillet and season them with salt and pepper. Cook over direct medium heat, with the lid closed as much as possible, until cooked as desired, 4 to 6 minutes for partially runny yolks. Transfer to plates and serve right away with the steak.