For the filling: Drain the peaches into a medium saucepan, reserving the peaches. In a bowl, whisk the brown sugar, cornstarch and cinnamon, then stir into the peach syrup, along with the vanilla extract.
Bring to a boil, stirring continuously, and cook until the syrup has thickened, about 5 minutes. Remove from heat and cool completely.
For the dough: Mix the flour and salt in a large bowl. Add the oil and stir to combine.
Add the milk, 1/4 cup at a time, mixing until the dough pulls away from sides of the bowl but is still a little wet.
Lay out the dough onto a lightly floured work surface and knead just until it comes together. Roll out to a 1/4-inch thickness and then cut out 6-inch circles, using a paring knife or an inverted bowl.
Place 1/4 cup of the filling in the center of each dough circle. Moisten the outer edges of the circles with water and fold the dough over to form half-moons. Crimp the edges with your fingers to seal.
Heat 1/2 inch oil in a large deep skillet over medium-high heat until it reaches 350 degrees F on a candy thermometer.
In batches of two, carefully place the pies in the oil and fry until golden brown, 6 to 8 minutes per side.
With a slotted spoon, transfer the fried pies to a paper-towel-lined plate to drain of excess oil, then immediately toss with cinnamon sugar to coat. Serve hot.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.