Preheat oven to 400 degrees F with racks in upper and lower thirds. Line two sheet pans with parchment paper. On a floured surface, unfold the puff pastry sheets and cut into each sheet into four pieces. Use a rolling pin to roll each piece to a 7in by 9in rectangle. Set on one of the prepared pans, layering between extra parchment as necessary and place in refrigerator to keep cool. Mix together chicken, pesto, and peas in a medium bowl. Place a heaping 1/2 cup of the chicken filling onto the bottom half of each pastry rectangle. Leave a small border on the bottom. Fold the top half of the pastry over the filing, lining up the edges. Press the edges with a fork to seal the turnover. Brush each one with beaten egg and sprinkle with Parmesan cheese and pepper. Bake the turnovers, switching the pans from upper to lower and lower to upper halfway through cooking, until pastry is deep golden brown and turnovers are starting to bubble at the seams, 25-30 minutes. Serve hot.