Using a food processor, pulse the flour, shortening and 1 teaspoon salt until the mixture resembles sand. With the machine on, add 1 cup cold water in a thin stream until the dough forms a ball. Shape into 2 equal rounds. Cover with plastic wrap and refrigerate for 30 minutes.
In a large skillet, heat the olive oil with 1/2 cup water over high heat. Add the onions and cook, stirring, until the water evaporates, about 10 minutes. Lower the heat and simmer, stirring, until the onions are golden-brown, about 30 minutes. Remove from the heat; season with salt and pepper and stir in the dates and balsamic vinegar. Let cool.
Preheat the oven to 400 degrees . On a floured work surface, cut each dough round into 6 equal pieces. Using a rolling pin, roll each piece into a thin round; trim using a 6-inch plate as a guide.
Beat the egg with 1 teaspoon water. Divide the onion mixture among the rounds and top each with 1 teaspoon cheese. Brush the edges with some of the egg wash, fold to enclose the filling and pinch the edges together to seal.
Divide the empanadas among 2 parchment-paper-lined baking sheets. Brush with the remaining egg wash. Bake until golden-brown, 20 to 25 minutes. Let cool for 15 minutes.