In a large glass mixing bowl, combine the finely diced fresh peaches, granulated sugar, ground cinnamon, fresh lemon juice, and all-purpose flour.

Stir vigorously to ensure the fruit is evenly coated. Let the mixture sit undisturbed for 10 minutes. The sugar will osmotically draw out the peach juices, which will immediately bind with the flour to create a thick, stable slurry. (This critical structural step prevents a watery filling from blowing out the pastry seams during the fry).

Unroll the cold puff pastry dough onto a lightly floured work surface. (Keep the dough as cold as possible to preserve the butter lamination).

Using a sharp 4-inch round pastry or cookie cutter, punch out as many circles as possible (approximately 8 per sheet).

Place one round of pastry dough flat on the board. Use a slotted spoon to scoop 1 to 2 tablespoons of the thickened peach filling precisely into the absolute center, leaving a strict ½-inch clean border around the edge. (Leave excess liquid in the bowl).

Place a second pastry round directly over the top of the filling. Using the tines of a silver fork, press down firmly all the way around the perimeter to aggressively crimp and seal the edges together. Repeat to form 8 sealed pies.

In a deep, heavy-bottomed cast-iron pot or Dutch oven, pour enough vegetable oil to reach exactly ⅓ up the sides. Attach a deep-fry thermometer and heat the oil to a strict 375°F over medium-high heat.

Carefully and gently lower 2 hand pies at a time into the hot oil. They will immediately sink, puff dramatically, and float to the top.

Fry until the bottom side is deeply golden brown and shattered, about 1 to 2 minutes. Use a professional wire spider or slotted spoon to carefully flip the pies and fry the second side until golden, about 1 to 2 minutes more.

Transfer the screaming-hot fried pies to a heavy wire cooling rack set over paper towels to drain excess oil and set the crust.

While still warm, aggressively dust the tops with a heavy layer of confectioners' sugar and a final pinch of ground cinnamon. Serve immediately while the pastry is crisp and the fruit core is molten.