Once the filling has cooled, place the pizza dough on a clean, flat work surface that has been dusted with all-purpose flour. Roll out the dough into a large rectangular shape, about 12x10". Cut the rectangle in half lengthwise, then cut each strip in half again, leaving you with four long strips. Cut all the strips in half widthwise then cut each piece in half again, leaving you a total of 16 squares.
In a small mixing bowl, whisk up the egg with a splash of milk and set aside. On each piece of dough, place a heaping tablespoon of filling in the middle of each square, leaving a small border around the filling. Using a pastry brush or the back of a spoon, brush the border with the egg and milk mixture. Fold all the sides inward, one at a time towards the filling, as if you were closing a box. Once the filling is enclosed, slightly pinch the seam closed. Transfer to a non-stick baking sheet, seam side down and repeat this process with the remaining squares of dough.
In a medium-size mixing bowl, combine the remaining Parmigiano Reggiano, oregano, basil and parsley. Mix to thoroughly combine. Brush the tops of each chicken parm bite with the remaining egg mixture and sprinkle the cheese-herb mixture over each of them. Bake the bites until evenly golden brown, 12-15 minutes.