For the pie dough: Pulse the flour, cocoa powder, sugar, baking powder and salt in a food processor until combined. Add the butter and shortening and pulse until the mixture looks like coarse meal. Drizzle in 6 tablespoons ice water and pulse until the dough starts coming together, adding 1 tablespoon of ice water if needed.
Turn the dough out onto a piece of plastic wrap and lightly knead to bring together. The dough might be a little crumbly, but it will hydrate as it chills and comes together. Form into a disk, wrap in the plastic wrap and refrigerate until firm, about 1 1/2 hours. It should feel like very firm modeling clay.
For the filling: Melt the butter in a medium saucepan over medium heat. Add the cherries and sprinkle with the sugar, cornstarch and salt and stir to combine. Cook, stirring constantly, until some of the cherries begin to break down and a jelly-like film begins to form at the bottom of the pan, 4 to 5 minutes. Use the back of a wooden spoon or rubber spatula to smash half of the cherries against the side of the pan. Remove from the heat and stir in the sour cherry jam and almond extract, if using. Transfer the filling to a medium bowl and let cool at room temperature for 15 minutes. Then transfer to the refrigerator until fully chilled, about 1 hour. Fold in the chocolate chips once chilled.
To assemble and fry: Remove the dough from the refrigerator and divide into 8 even pieces, then roll each piece into a ball. On a lightly floured work surface, roll out each ball into a 6 1/2-inch round. If the dough starts to crack in places as you roll it out, pinch and press it together with your fingers and re-roll until the dough is smooth again. Stack the rounds between pieces of parchment paper and refrigerate until firm, about 30 minutes.
Remove the dough from the refrigerator and let sit for 5 minutes. Beat the egg in a small bowl with 1 tablespoon of water.
Heat 1 inch of vegetable oil in a Dutch oven or deep high-sided skillet until a deep-fry thermometer registers 375 degrees F.
Scoop about 3 tablespoons of the cherry filling into the center of each dough round. Brush the edge on one side of the dough round with some egg wash and fold the dough over to make a half-moon shape. Press out the air and seal the edges with your fingers. Use a knife to trim any of the rough edges away, leaving 1/2 inch of the dough around the filling. Crimp the edges with a fork. Keep the formed hand pies on their individual pieces of parchment as you form them to make them easier to handle when adding to the hot oil.
One at a time, peel away the parchment paper and add 4 of the pies to the hot oil, turning once, until the dough is bubbling in spots and crispy, about 4 minutes. Transfer the pies to a wire rack set over a baking sheet and immediately sprinkle generously on all sides with sugar.
Repeat with the remaining pies. Allow the pies to cool for 10 minutes, then serve warm.