- In a bowl, combine the beaten egg and milk. Set aside.
- Line a baking sheet with parchment paper.
- On a floured work surface, roll out the dough into a 30 x 25-cm (12 x 10-inch) rectangle. Cut into three 10-cm (4-inch) wide strips. Brush the first strip with the classic pesto, the second with the sundried tomato pesto and the third with the beaten egg mixture sprinkled with the Parmesan.
- Fold each dough strip, starting at each edge and roll inward 3 times, about 1.5-cm (1/2-inch) at a time. Brush the sides of each piece of dough with the beaten egg mixture, then fold in half. Place on the baking sheet and freeze for 45 minutes.
- Meanwhile, with the rack in the middle position, preheat the oven to 180 °C (350 °F).
- Cut each piece of dough into ½-cm (1/5-inch) thick slices, about 12 palmiers per piece of dough. Place the palmiers on the baking sheet. Bake for 12 to 13 minutes or until the palmiers are deep golden brown. If desired, freeze the palmiers, thaw at room temperature and reheat in the oven before serving.