For the pie dough: Combine the flour, sugar and salt in a food processor and pulse to combine. Add the shortening and butter and pulse until the mixture resembles coarse breadcrumbs. Add in the ice water and vinegar, 1 tablespoon at a time, and pulse until the mixture just comes together. Turn the dough out onto a large piece of plastic wrap and press into a large disc. Refrigerate for at least 1 hour.
For the filling: Melt the butter in a large saute pan over medium-high heat and then saute the apples, stirring, until just starting to soften, 3 to 5 minutes. Sprinkle with the sugar and stir. Add the cinnamon, salt, ginger and nutmeg and cook, stirring, until some of the apples start to break apart (but most are still whole), about 4 minutes. Stir in the lemon juice and remove from the heat. Set aside to cool completely, about 30 minutes.
Fill a deep pot halfway with oil and heat to 365 degrees F. Roll the dough out 1/8-inch thick on a lightly-floured surface. Cut out eighteen 4 1/2-inch-diameter circles, gathering and re-rolling the dough if necessary. Beat the egg in a small bowl with 1 tablespoon of water. Place about 1 heaping tablespoon of the apple filling into the center of each dough round. Brush the edge with some egg mixture and fold the dough over to make a half-moon shape. Press out the air and seal the edges with a fork. Repeat with the remaining filling and dough.
Place 4 to 6 pies at a time into the hot oil and cook until golden brown, flipping as needed, about 5 minutes. Transfer to a paper towel-lined plate. Serve warm or at room temperature.
Cook's Note: The pies can be frozen for up to 6 months before frying. Frozen pies will take about 8 minutes to fry. You can also refrigerate the pies for up to 2 hours before frying.