Preheat the oven to 400°F, with racks in the upper and lower thirds. Line two sheet pans with parchment paper. 

On a well-floured surface, carefully unfold the puff pastry sheets. Use a sharp knife to cut each into four pieces. Use a floured rolling pin to roll each piece into a 7-by-9-inch rectangle. Set the pastry rectangles on one of the prepared pans, layering them between pieces of parchment if necessary. Place the puff pastry in the refrigerator to chill while you mix up the filling. 

Mix together the chicken, pesto, and peas in a medium-size bowl. Place a heaping ½ cup of filling onto the bottom half of each pastry rectangle, leaving a small border at the bottom. Carefully fold the top half of the pastry over the filling, lining up the edges. Press the edges with the tines of a well-floured fork to seal. Arrange the turnovers on the prepared pans, spacing them evenly apart.

Brush each turnover with beaten egg and sprinkle with Parmesan cheese and black pepper. 

Bake the turnovers, switching the pans from upper to lower and lower to upper halfway through cooking, until the pastry is a deep golden brown and the turnovers are starting to bubble at the seams, 25 to 30 minutes. Enjoy the turnovers hot from the oven.