Preheat oven to 400ºF with racks positioned in the upper and lower thirds of the oven. Line 2 baking sheets with parchment paper.

Roll out puff pastry sheets on a floured work surface to almost double in size;  cut each into 4 pieces. Stack between parchment paper and refrigerate for 10 minutes.

Combine basil, chicken, and green onions. Arrange 4 pieces of chilled puff pastry on a floured work surface; divide chicken mixture among sheets, piling it on the bottom half of each rectangle, leaving about ⅓ inch bare on bottom edge and along sides. Top evenly with sliced mozzarella, fold the top half over the filling, matching up edges, and seal carefully with the floured tines of a fork, transferring to prepared baking sheets as you go.

Combine the egg and cream and brush the tops of each pocket with the mixture. Sprinkle each with Parmesan cheese and cut 3 slits into the top of each. Transfer to the oven and bake 15 minutes. Switch the pans between the 2 racks, rotating front to back as you do. Continue to bake until golden brown, 10-15 minutes. Serve hot.