Make dough. In a large mixing bowl, whisk together the flour, sugar, and salt. Then, add the cold, cubed butter to the bowl. Blend the butter into the flour mixture using a pastry cutter or your fingertips until it resembles coarse crumbs. Gradually sprinkle ice water into the bowl, one tablespoon at a time, mixing with a fork until the dough just begins to come together. When the dough holds together without being wet or sticky, stop adding water. Press the dough into a ball, then flatten it into a disk.
Chill. Wrap the dough disk tightly in plastic wrap. Refrigerate for 30 minutes to allow the gluten to relax, making the dough easier to roll out.
Make filling (optional). (Note: I almost always use canned pie filling.) In a medium bowl, combine the blueberries, sugar, cornstarch, lemon zest, and lemon juice. Gently toss the ingredients together until the blueberries are coated evenly. Set aside to marinate while the dough chills.
Roll and cut. Remove the chilled dough from the refrigerator and let it sit on the counter for a few minutes to soften slightly, making it easier to roll. Lightly flour your work surface and rolling pin. Roll the dough out to a thickness of about ⅛-inch. Use a round cookie cutter or the rim of a glass to cut out 4-inch circles. Re-roll any scraps and continue cutting until all the dough is used.
Assemble. Place about 1 tablespoon of the blueberry filling in the center of each dough circle. Be careful not to overfill. Fold the dough over the filling to form a half-moon shape. Press the edges together gently and seal by crimping with a fork.
Fry. In a deep skillet or heavy-bottomed pot, heat the oil to 350°F (175°C). Use a thermometer to check the temperature for safety and accuracy. Fry the hand pies in batches, careful not to overcrowd the pan, until golden brown and crispy, about 3-4 minutes per side. Use a slotted spoon or tongs to transfer the pies to a paper towel-lined plate to drain.
Glaze. While the pies cool slightly, whisk together the powdered sugar, milk, and vanilla extract in a small bowl until smooth. The glaze should be thick enough to coat the back of a spoon. Dip each warm hand pie into the glaze, making sure both sides are well coated, then set on a wire rack to dry.
Serve. Serve the hand pies warm for the best flavor and texture.