To make the dough: in a food processor, mix the flour and salt together. Add the butter, egg, and 5 tablespoons of water until a clumpy dough forms. Be careful not to overwork the dough. Just mix until it comes together.
Form the dough into ball, wrap in plastic wrapper, and chill in the refrigerator for approximately 30 minutes. Then, unwrap and roll out the dough into a thin sheet over a floured surface, working from the center of the dough to the edges.
Cut out round disc shapes using a round medium-sized or small mold or cutter. Use immediately or store in the freezer for up to 3 months. (In the latter case, pile one disc on top of the other with plastic wrap in between them. Wrap the pile well and store in a ziploc bag. Thaw in the refrigerator before filling the pizza empanadas).
For the filling: In a small bowl, mix both the ham and cheese. Reserve.
Assemble the pizza empanadas, adding a spoonful of the ham and cheese mixture on the center of each empanada disc. Top with 2 halves of a tomato and sprinkle a little bit of oregano.
Brush the edges of the pizza empanada discs with the egg whites or a little bit of water (although the whites tend to be more effective at sealing the edges of the pizza empanadas).
Fold the empanada discs and seal the edges gently with your fingers. Then, press the edges of the pizza empanadas with a fork to crimp. Lightly brush the top of the pizza empanadas with the egg yolk thinned with 1/2 teaspoon of water.
Layer the pizza empanadas in a single layer in a deep baking pan or pyrex dish. Cover with aluminum foil (don't let it touch the pizza empanadas). Let the pizza empanadas rest in the fridge for about 30 minutes or until ready to bake.
Meanwhile, pre-heat the oven to 400 degrees F (200 degrees C) and bake empanadas for 15 - 20 minutes, or until golden. Serve pizza empanadas warm by themselves or with a dipping sauce (e.g. chimichurri).