Preheat oven to 400 degrees F.
On a lightly floured surface, roll out puff pastry to a 10 by 10-inch square. Spread 3 tablespoons of tomato paste evenly on top, followed by 3 tablespoons of olive tapenade. Sprinkle half of mozzarella and Parmesan over top. Roll up the puff pastry like a jelly roll, just to the middle of the dough. Repeat with the other side in the same fashion, making 2 rolls that meet in the center. Duplicate the process with the second puff pastry sheet.
Line a cookie sheet with parchment paper. Slice the rolls into 1/2-inch slices. Arrange on cookie sheet and refrigerate for 20 minutes. (Because they will puff up to twice their size after baking, place them far enough apart.) Just before baking, beat the egg with 2 tablespoons water in a small bowl, and brush onto the olive crisp slices. Bake for 20 minutes, or until puffed and golden. Serve warm or room temperature.