Preheat the oven to 400°F with the rack set in the lower third of the oven. Lightly butter a 17 x 14-inch baking sheet.
Roll the pastry into a rectangle ⅛inch or less thick, and large enough to cover the baking sheet. Sprinkle the baking sheet with water. Lift the pastry onto a rolling pin and unroll it onto the baking sheet, pressing down gently. Trim the edges of the pastry even with the edges of the baking sheet. Brush the pastry with the beaten egg and place in the refrigerator for 15 minutes.
Cut the chilled pastry into strips 3½inches long and 1inch wide, for a total of 64. The pastry can be made ahead to this point. Refrigerate or freeze on the baking sheet. Bake without defrosting, adding 1 minute to the cooking time.
Bake the pastry strips for 20 to 25 minutes, or until the pastry has risen and browned evenly. Transfer to a rack, but while still warm, split the strips in half horizontally. (The strips can be baked in advance and reheated in a 350°F oven for 3 minutes before serving.)
Make the Anchovy Butter: Cream the butter with a whisk. Add the minced anchovy fillets, lemon juice, mustard, and pepper and blend well.
Spread the bottoms of the pastry strips evenly with the anchovy butter, replace the tops, and serve lukewarm.