Season tournedos well on all sides with salt and pepper. Heat a small skillet over medium-high heat until hot. Add the vegetable oil and the butter. Add the tournedos to pan and cook for 2 minutes on each side for medium-rare. Serve two tournedos per person, topped with the Bearnaise Sauce.†
In a saucepan combine shallots, lemon juice,vinegar and tarragon stems and cook over medium heat until liquid has reduced to 2 tablespoons. Let reduction cool to room temperature.
In a metal bowl, combine egg yolk, water and reduction over a pot of simmering water. Whisk well until egg mixture begins to form ribbons when whisk is lifted out of mixture. While continuing to whisk, drizzle in melted butter little by little until completely incorporated and sauce is thick. Add chopped tarragon, taste and season with salt and white pepper.
Yield: sauce for 2 people, about 1/2 cup