Preheat oven to 350°F with racks in upper and lower thirds of oven. Line two baking sheets with parchment paper; set aside.
Heat oil in a medium skillet over medium-high. Add onion and salt; cook, stirring often, until onion is translucent and starting to brown, about 6 minutes. Add garlic, and cook, stirring constantly, until fragrant, about 1 minute. Remove from heat, and transfer to a medium bowl. Stir in spinach and queso Oaxaca.
Unroll 1 piecrust on a lightly floured surface. Using a 3 1/2-in. round biscuit cutter, cut out rounds, and place on the two prepared baking sheets. Reroll dough scraps to get a total of 10 rounds. Repeat the procedure with remaining piecrust.
Fill each dough round with 1 heaping tablespoon of spinach mixture. Fold one half of dough over filling, and press to adhere to other side of dough. Crimp edges with a fork to seal.
Whisk egg and water in a small bowl; brush over empanadas. Bake in preheated oven until golden brown, 30 to 35 minutes, rotating and switching pans to opposite racks halfway through cooking. Remove and set aside to cool slightly, about 10 minutes. Serve warm.