Using a round pastry cutter, cut the pastry into 2-inch rounds. Roll the remaining scrap into a ball and roll out with a rolling pin into a 1/4-inch thick sheet and continue to cut rounds of puff pastry until you've used up all of the pastry. As you cut the pastry, place it on a cookie sheet lined with waxed paper and refrigerate it until the filling has been made.
For the filling, put the remaining ingredients, except the egg wash, into a food processor and pulse for 1 to 2 minutes to combine well. Scrape the sides of the bowl carefull with a rubber spatula and pulse again briefly.
Heat a deep fat fryer to 365 degrees.
Lightly brush a round of pastry with some of the egg wash. Place 1 teaspoon of the ham filling in the center of the round and fold the dough in half to create a dumpling. Continue to fill the pastry in this way until all of the filling and pastry has been used up. Place the finished turnovers on a parchment paper-lined cookie sheet, brush the tops with the remaining egg wash, and deep fry in batches for several minutes until rich golden brown. Drain on paper towels and serve immediately.