Preheat oven to 400°F (200°C). Line a baking sheet with parchment.
Roll out puff pastry on lightly floured surface into a 10 x 8-inch rectangle.
Spread pesto evenly, leaving a 1/4-inch border. Layer mozzarella slices, sprinkle Parmesan, season lightly.
Fold long edges toward center so they meet; fold one side over the other to form a log. Chill 10 minutes.
Slice log into 1/2-inch pieces, place cut-side up on sheet. Brush with egg wash.
Bake 12–15 minutes until golden and puffed. Cool slightly before serving.