Artichoke Preparation: Cut the stems and most of the leaves off the artichoke. With a paring knife, whittle off the leaves at the base to expose the heart. Scrape the fibrous "choke" to leave only the disk-shaped heart. Rub the heart with lemon. Cook with salted boiling water for about 8 minutes. Drain and refresh with cold water. Set aside.
Bearnaise Sauce: In a heavy saucepan, combine wine, tarragon vinegar, shallots, fresh tarragon and white pepper. Boil to reduce to 3 tablespoons. Let cool. Meanwhile, melt the butter; cool slightly. Add egg yolks to vinegar mixture and cook over low heat, whisking constantly, for 1 minute or until the consistancy of mayonnaise. Do not overcook or it will curdle. Whisk in the warm melted butter in a thin stream into the wine mixture. The sauce will thicken. Whisk in salt, cayenne and white pepper to taste. Keep warm in a pan of warm water.
Poach the 4 eggs 3 minutes in 2 quarts of water and 1 tablespoon wine vinegar. Remove with a slotted spoon and trim edges. Keep in warm water. To assemble, heat the artichoke hearts and blot dry. Spoon a bit of the Bearnaise Sauce into each heart. Place a poached egg on top and spoon more sauce over. Serve at once.