Place the puff pastry on a work surface and spread the mustard over the top. Arrange the prosciutto evenly over the mustard to cover all the pastry, and then sprinkle with the Parmesan. Lightly press the cheese into the prosciutto with a rolling pin.
Sstarting at one long edge, roll up the puff pastry like a jelly roll just to the middle of the dough; then roll up the other side in the same fashion, making two rolls that meet in the center.
Using a serrated knife, cut the rolls crosswise into 1/2" slices. Place the slices on cookie sheets lined with parchment paper and press lightly with your hands to flatten. Refrigerate for 15 minutes.
Preheat oven to 400 degrees.
Beat the egg and water together in a small bowl. Brush the top of each palmier with the egg wash. Bake until puffed and lightly golden, about 10 minutes. Serve warm or at room temperature.