Preheat oven to 400 degrees. Remove pizza dough from refrigerator and allow to rest on the counter while you prepare the filling.
Combine all ingredients in a bowl except for the egg. Roll the pizza dough onto a floured surface. Using a cereal bowl, trace four 6-8 inch circles in the dough. Place the filling on one half of each circle, leaving 1/2 inch of perimeter. Fold the other half of the dough over the filling and crimp the edges with a fork.
Place the pockets onto a parchment-lined baking sheet. Cut a tiny slit or two into the crust. Brush with the egg and sprinkle with additional Asiago cheese. Bake 22-25 minutes.