In a large bowl, stir together the flour, cornmeal, sugar and salt. Using a pastry cutter or fork, work the cold butter into the flour mixture until it is crumbly and the butter is pea-sized.
Shift the butter and flour mix to one side of the bowl. On the empty side, add 1 TB of water and toss the dry mix into it. Continue this process, one tablespoon of water at a time, until the mixture is moist.
Work quickly and do not overwork the mix or the crust will be tough.
Press the dough together into a flat disc, wrap tightly in plastic wrap and chill for at least one hour.
Preheat oven to 375 degrees and place oven rack in center. Line a baking sheet with parchment paper and set aside.
On a lightly floured surface, roll the prepared cornmeal crust until thin (like you would use for a traditional pie). Cut into even rectangles. (The crust recipe will make about 12 hand pies.)
On one side of each rectangle, spoon about 2 tablespoons of chopped Luck's apples. Fold the dough over, press the edges together, and crimp with a fork to seal.
Repeat with each piece of dough.
Whisk together the egg and 1 teaspoon of water. Brush the top of each pastry lightly with egg wash. Cut small vents into each hand pie. Sprinkle the top with brown sugar (if desired).
Bake for 40 minutes until the hand pies are golden. Transfer to a wire rack to cool.