03In a small saucepan, combine the water, vinegar, tarragon, shallots and white wine. Bring to a boil, and reduce until only about 2 tablespoons liquid remain in the pan. Remove from heat and let cool.04In the top bowl of a double boiler over medium-low heat, whisk together the egg yolks and the reduced tarragon liquid. Remove from heat, and whip the butter very gradually into the mixture. When all the butter has been absorbed, season with salt, pepper and lemon juice. Keep warm until served.05In a deep saucepan or wok, heat vegetable shortening to 375 degrees. There should be enough oil to cover the artichokes. Using up to six artichoke halves at a time, dip each in the batter, drain the excess and drop in the oil, frying until golden brown. Remove and drain well on paper towels, and keep warm. Repeat with remaining artichokes.06Serve with bearnaise sauce.