In a small saucepan, combine vinegar, wine, 2 tablespoons tarragon, shallots, and peppercorns. Bring to a boil over medium-high heat and reduce mixture to 2 tablespoons, about 7 minutes. Strain liquid and discard solids. Cool slightly.
Place egg yolks in a metal bowl over a double boiler, add vinegar reduction, and whisk for 2 minutes. While whisking, slowly pour in melted butter in a slow and steady stream. Add remaining 2 tablespoons of tarragon and 1 teaspoon salt, and whisk rapidly for another 45 to 60 seconds, until thickened. If sauce is too thick, add 1 tablespoon very hot tap water. (The bearnaise sauce can be kept at room temperature for an hour before serving.)
Season steaks on both sides with the remaining 1 tablespoon of salt and the black pepper.
Preheat the oven to 400 degrees.
Heat a large grill pan over medium heat and add olive oil. When oil begins to smoke, add steaks and cook for 10 to 12 minutes per side, depending on thickness, for medium-rare. Remove steaks from grill and allow to rest for 8 to 10 minutes before slicing thinly across the grain.
Serve steak with pommes frites and bearnaise sauce on the side.