In a mixing bowl, combine the beef, flour and paprika. Stir to combine. Add the thyme, garlic, potato, onion and some salt and pepper. Stir to combine.
Divide one portion of the chilled Part-Butter/Part-Shortening Easy Pie Dough into 8 equal pieces. Roll each piece of dough into a round about 3 inches in diameter. You can use a 3-inch round cookie cutter to cut each round and even the edges if you desire. Repeat with the second piece of chilled dough.
Preheat the oven to 350 degrees F.
Brush each round of dough with the egg wash. Divide the filling among the rounds, piling it in the middle of each round. Bring the sides of the dough up to meet the middle and gently crimp the edges down in the center. Cut 3 small slits into each hand pie to allow steam to escape. Brush the top of the pies with egg wash.
Bake until golden brown, about 30 minutes. Allow to cool slightly before serving.
Part-Butter/Part-Shortening Easy Pie Dough:
In the bowl of a food processor fitted with the blade attachment, pulse together the flour, sugar, salt, butter and shortening until the mixture resembles cornmeal but there are still pea-sized chunks of fat.
In a small bowl, stir together the ice water and lemon juice. Slowly add the liquid to the flour mixture, pulsing, until the dough just comes together. Squeeze a small piece of dough between your thumb and index finger to make sure it holds its shape.
Turn the dough out onto a lightly floured surface and divide it in half. Gently turn over each piece of dough a few times so that any dry bits are incorporated. Form each piece into a loose disk, cover the dough with plastic wrap and let it rest in the refrigerator for at least 20 minutes.