Bring a medium saucepan of lightly salted water to a boil and add the fennel. Cook for 1 minute. Drain, rinse under cold water, and drain again. Place in a bowl and add the Tomato Confit, olives, and capers. Mix well and set aside.
In a small bowl, beat the egg white until foamy. In another small bowl, beat the egg yolk and heavy cream. Set the two bowls aside.
Fold each piece of parchment paper in half. Using scissors, cut each folded piece of paper into a wide, plump heart shape (each opened heart should be about 12 inches across its widest point). Place a opened parchment heart on your work surface, with the fold running vertically. Leaving a 1-inch border, spread one side of the heart with 1 tablespoon anchovy butter. Top with 1 turbot fillet, one-fourth of the fennel mixture, 2 tablespoons vermouth, and 1 additional tablespoon of anchovy butter. Season with the salt and pepper mixture. Brush the outside edge of the heart with the beaten egg white. Fold the unbuttered side of the parchment heart over to cover the fish. Using tight interlocking folds, close the edges of the paper heart. Place on a baking sheet. Repeat with the remaining ingredients. Brush the tops of the papillotes with the egg yolk mixture. Refrigerate for 1 hour.
Preheat the oven to 375 degrees F. You will need 1 large skillet for each papillote. Place 1 tablespoons of oil in each skillet, and heat over medium heat. Add 1 papillote to each skillet and cook for 3 to 4 minutes, until the undersides are deeply browned. Transfer the skillets to the oven and bake until the top of the papillate is browned, about 5 minutes.
To serve, cut open each papillote and transfer the contents, including all the juices, to warmed dinner plates
Rinse the anchovies well to remove their salt. Run your thumbnail along the backbone of each anchovy to remove the fillets. Discard the bones and mince the fillets into a paste. Transfer the paste to a small bowl and mix in the remaining ingredients. (The anchovy butter can be prepared up to 1 day in advance, covered and refrigerated. Remove from the refrigerator and let stand until softened before using.);