Heat oven to 400° F. In a small bowl, cover tomatoes with warm water and rehydrate for 10 minutes; remove and pat dry with paper towels. Add to food processor with artichokes, olives, Parmesan and pepper. Pulse to combine until finely chopped.
Gently roll out one pastry sheet on parchment paper. Spread half the mixture evenly on top. Fold the short ends of the dough over about 2 inches; fold each side again until the edges almost touch. Fold one more time, bringing one side on top of the other and press lightly with the rolling pin. Repeat with second sheet. Cover both with plastic wrap and refrigerate 30 minutes.
Beat 1 egg with 1 tablespoon water in a small bowl. Remove pastries from refrigerator and brush with the egg mixture. Slice each roll into 20 pieces and divide among four parchment paper lined baking sheets, leaving a 2-inch space between each puff. Bake at 400° F for 12 to 15 minutes or until golden.