Step 1

For the dough: Add the flour, sugar and salt to a food processor and pulse to combine. Add the butter and pulse until mealy. Combine the water and white vinegar in a liquid measuring cup; slowly drizzle into the flour mixture with the food processor running until the dough begins to become homogeneous—do not overmix. Turn the dough out onto a clean surface and press into a disc. Wrap tightly with plastic wrap and refrigerate for 30 minutes.

Step 2

For the filling: Add pastrami, sauerkraut, cheese, mustard, apple cider vinegar and salt to a large bowl and toss to combine. Set aside.

Step 3

For the assembly and frying: Divide the dough into 10 even pieces. Roll each piece into a round 5 inches in diameter and 1/4 inch thick. Keep the rounds covered with plastic wrap as you work with 1 round at a time. Whisk the egg, salt and 1 tablespoon water. Brush around the edge of a dough round and add about 2 tablespoons of filling to one side of the round, leaving 1/2 inch of the dough empty at the edge. Fold the round in half over the filling to create a half-moon shape, crimp the edges with a fork to seal then trim off any excess dough.

Step 4

Fit a Dutch oven or large pot with a deep-fry thermometer, fill halfway with oil and heat over medium-high heat until the oil reaches 350 degrees F. Using a spider or slotted spoon, gently add 1 or 2 pies at a time and fry, flipping once, until golden brown, 5 to 6 minutes. Transfer the pies to a wire rack to cool slightly.