How to Make It
Light a grill or heat a grill pan. Season the steak with salt and pepper and grill over high heat, just until nicely charred outside and raw within, about 1 1/2 minutes per side. Let the steak cool, then refrigerate until thoroughly chilled, about 1 hour.
Meanwhile, in a small saucepan, combine the wine vinegar with half of the minced shallots and boil over high heat until the vinegar is reduced to 1 tablespoon, about 3 minutes. Transfer the reduction to a small stainless steel bowl.
In a small saucepan, bring 1/2 inch of water to a boil; keep the water simmering over low heat. Add the egg yolks to the vinegar reduction and whisk over the simmering water until the mixture is thick, about 1 minute; if the egg sticks to the sides, remove the bowl from the simmering water to prevent overcooking the yolks.
Remove the bowl from the simmering water and gradually whisk in the melted butter. Stir in the chopped tarragon and season with salt and pepper.
Trim the fat and silver skin from the steak. Chop the meat very finely and transfer to a bowl; alternatively, pass the meat through a meat grinder. Stir in the parsley, chives, olive oil and the remaining shallots; season lightly with salt and pepper.
Mound the steak tartare in the center of each plate. Carefully reheat the béarnaise sauce over the pan of simmering water, whisking constantly. Spoon the béarnaise sauce over the steak tartare and serve right away, with potato chips.