Combine the black pepper and steak seasoning in a small bowl and set aside. Combine the vinegar, water, shallot, and 2 tablespoons of the tarragon in a saucepan over medium-high heat. Bring to a boil and let the liquid reduce to a couple of tablespoons. Remove the pan from the heat and let the vinegar mixture cool. Melt the butter in a saucepan over medium heat. Do not let it burn but it should be hot. Combine the egg yolks, salt, cayenne, and cooled vinegar mixture in a blender. Process until combined. Slowly add the hot butter through the feed tube in the blender, processing until thick. Add the remaining tarragon and process for 10 seconds. Meanwhile, heat the oil in a skillet over medium-high heat. Press 1/2 teaspoon of the seasoning mixture on each side of the steaks. Add the steaks to the hot skillet and cook for 3 minutes per side or until desired doneness. Serve the peppery steaks hot, topped with some of the bearnaise sauce.