In a large bowl combine masa harina, flour, baking powder, and salt. Using a pastry blender, cut in shortening until mixture resembles coarse crumbs. Gradually add the cold water to the flour mixture, stirring to make a dough.
Divide the dough into 12 portions; shape each portion into a ball. Place each dough portion between two pieces of waxed paper. Using a tortilla press or rolling pin, flatten each ball into a 6-inch circle.
Preheat oven to 200 degrees F. In a medium bowl combine chicken, cheese, the 1/2 cup salsa verde, and the oregano. Remove the top sheet of waxed paper from one dough circle. Spoon a generous tablespoon of the chicken mixture into the center of the dough circle. Using the waxed paper as a guide, fold over the dough into a turnover shape; seal dough edges by pressing with your fingers or the tines of a fork. Repeat with the remaining dough circles and chicken mixture to make 12 empanadas total.
Carefully peel each empanada off the waxed paper. In a large skillet heat 1 inch of vegetable oil over medium-high heat. Fry empanadas, a few at a time, for 3 minutes or until golden brown, turning once halfway through frying time. Drain on paper towels. Transfer empanadas to a baking sheet and keep warm in the preheated oven while frying remaining empanadas. Serve empanadas warm with additional salsa verde.