For the sauce, put the Champagne vinegar, white wine, shallots, 1 tablespoon tarragon leaves, 1/4 teaspoon salt, and 1/4 teaspoon pepper in a small saucepan. Bring to a boil and simmer over medium heat for about 5 minutes, until the mixture is reduced to a few tablespoons. Cool slightly.
Place the cooled mixture with the egg yolks and 1 teaspoon salt in the jar of a blender and blend for 30 seconds. With blender on, slowly pour the hot butter through the opening in the lid. Add the remaining 2 tablespoons of tarragon leaves and blend only for a second. If the sauce is too thick, add a tablespoon of white wine to thin. Keep at room temperature until serving.
Season the steaks liberally with salt and coarsely ground black pepper on both sides. Heat a thin layer of olive oil in a large saute pan over high heat until it's almost smoking, then sear the steaks on each side for 1 minute. Lower the heat to low and cook the steaks for about 7 to 10 minutes, turning once, until very rare in the middle. Remove to a plate, cover tightly with aluminum foil and allow to sit for 10 minutes. Serve with the bernaise sauce on the side.
Note: To make the sauce in advance, prepare an hour before serving and allow it to sit in the blender. Before serving, add 1 tablespoon of the hottest tap water and blend for a few seconds.
Consumption of raw or undercooked eggs, shellfish and meat may increase the risk of foodborne illness.