In a large glass mixing bowl, combine the finely diced fresh peaches, cane 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 draw out the natural peach juices, which will immediately bind with the flour to create a thick, stable slurry. (This critical 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 onto your work surface. Use a slotted spoon to place 1 to 2 spoonfuls of peach filling into the center of the dough. Leave a border around the edge. (You want the peaches, not much of the liquid.) Take another round of dough and place it on the top of the filling. Use a fork to crimp the edges together. Repeat with the remaining rounds.

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° over medium-high heat.

Gently place the pies 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 slotted spoon or wire basket to carefully flip the pies to cook until golden on the second side.

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.