Place the flour, salt, and butter in a bowl. Mix by hand and pinch the butter pieces with the flour until the mixture resembles coarse meal.
In a separate bowl, combine the egg, 1/3 cup cold water, and vinegar. Mix just enough to break up the egg.
Add the liquids to the flour. Mix by hand until the dough just comes together. Knead the dough 4 to 5 times. Form the dough into a flat disc. Wrap and refrigerate for at least 1 hour. This can be made up to 2 days ahead of time.
To make the peach filling, begin by dissolving the cornstarch in 3/4 cup of room temperature water.
Cook the sugar, vanilla seeds, peaches, and nutmeg in a saucepan over low heat for 3 to 4 minutes, or until the sugar is dissolved.
Increase the heat to high. Bring the sauce to a boil and add the cornstarch-water mixture. Bring it back up to a boil and reduce heat to medium, stirring continuously for 2 more minutes. Remove from the heat and let cool to room temperature, about 30 minutes.
Remove the dough from the refrigerator and roll out on a well-floured surface to no thicker than 1/4-inch. Using a knife or small inverted bowl, cut the dough into 4-inch circles. Dust off any excess flour.
Place a heaping spoonful of the cold filling in the middle of each pie. Moisten the edges of the pies with water and fold over the filling, pinching the edges together to remove excess air that may be trapped inside the pie.
Using a fork, gently press the edges of the pies to form a decorative pattern. The pies can sit in the refrigerator for up to 20 to 30 minutes before frying. They will become soggy if they sit for any longer.
To fry, heat 2 inches of oil in a large cast iron pan and use a candy thermometer to heat to 350 degrees F. Or use a deep fryer if you have one at home.
Using a spider and working in batches, carefully lower a couple of pies at a time into the oil, frying each side for about 2 minutes. Carefully remove the pies from the oil and place them on a stack of paper towels to blot any excess oil. The just-fried pies can be transferred to a baking sheet and kept warm in a 250 degree F oven while the others are frying. Toss them in cinnamon sugar and serve!