In a food processor, pulse the flour, salt and oregano together.
Dot the butter pieces evenly over the flour, then pulse several times until the butter pieces are about the size of little pebbles.
Drizzle two tablespoons of water evenly over the mixture, then pulse a few times. Drizzle in one more tablespoon of water and pulse just until the dough starts to clump together. (If it looks too dry, repeat with another tablespoon of water.)
Dump the dough out onto a lightly floured work surface and shape it into a flat disc. Wrap in plastic wrap and chill in the fridge for about an hour.
Once the dough has chilled, flour your work surface (or use a silicone mat or parchment paper) and roll out your dough until it’s about ⅛-inch thick. Using a 2-inch or 1.5-inch cookie cutter (depending on how many you want) cut out as many tops and bottoms as you can.
Place the cutouts on a parchment-lined baking sheet, cover and chill them for just a few minutes to firm up.
On half of the cutouts, use your beaten egg to paint on a border. These are your bottom crusts.
Assemble the pie pockets in this order: bottom crust, a teaspoon of pizza sauce, 3 pieces of mozzarella (or about a pinch-full if shredded), 4 pepperoni halves, then a pinch-full of basil strips. (For smaller pies, use a half-teapoon pizza sauce, 2 pieces of mozzarella and 3 pepperoni halves.)
Stretch out the top crust just a little bit and place on top to seal off all the ingredients. Use a fork to crimp the edges.
Brush on the egg wash over each pocket, then place them in the freezer for 10 minutes.
Preheat the oven to 375°F.
With a small knife, poke two vents in each pocket, then brush with egg wash a second time. Sprinkle with parmesan cheese, if desired.
Bake the pizza pie pockets for 16-18 minutes. Serve with leftover pizza sauce as a dip!