Melt the butter in a medium saucepan over medium heat. Add the frozen peaches, brown sugar, granulated sugar, cinnamon, cinnamon paste, vanilla bean paste, lemon extract, kosher salt, nutmeg, and cloves, if using. Stir gently until the peaches begin to soften and release their juices.

Continue cooking, stirring occasionally, until the peaches are completely thawed and tender enough to break apart with the back of a spoon. Gently break the larger pieces into small, bite-sized chunks.

Once the peaches have released their juices and a syrup has formed, reduce the heat to low. Mix the cornstarch with a small amount of cold water to create a slurry, then stir it into the peaches. Cook briefly until the filling thickens to a glossy, jam-like consistency with very little loose liquid. Remove from the heat and allow to cool completely.

Remove the biscuits from the refrigerator and let them sit at room temperature for a few minutes to make them easier to handle. Separate the biscuits and gently stretch each one into a slightly wider round, being careful not to tear the dough.

Working with four biscuits for the first batch, place approximately 1 tablespoon of cooled peach filling in the center of each biscuit. Keep the filling piled toward the middle, leaving a clean border around the edges. Gently stretch the remaining four biscuits and place one over each filled biscuit.

Press the edges together firmly with your fingers, then crimp with a fork to seal. For a decorative rope edge, gently pull, roll, and tuck small sections of dough around the perimeter until completely sealed.

Transfer the assembled biscuit hand pies to a parchment-lined tray and refrigerate while preparing the pie crust version.

Unroll the refrigerated pie crust onto a lightly floured surface. Cut into rounds large enough to hold approximately 1 tablespoon of peach filling while leaving room to seal. Gather and reroll the scraps as needed.

Spoon the cooled filling onto one half of each round, keeping it away from the edges. Fold the dough over to create a half-moon shape, gently push the filling toward the center, and press the edges together. Crimp with a fork to seal completely. Refrigerate until ready to fry.

Pour approximately 2 inches of neutral oil into a heavy-bottomed skillet or Dutch oven and heat to about 325°F. Working in batches, carefully lower the biscuit hand pies into the hot oil without overcrowding the pan.

Fry for approximately 4 minutes total, turning after the first 60 seconds and continuing to flip as needed until the pies are puffed, evenly golden brown, and cooked through. Adjust the heat as necessary to prevent the exterior from browning too quickly. Transfer to a wire rack to drain.

Fry the pie crust hand pies in batches, turning after approximately 60 seconds and continuing until both sides are golden brown and the pastry is cooked through. These may require fewer turns than the biscuit version. Transfer to a wire rack.

To prepare the glaze, whisk together the powdered sugar, vanilla bean paste, and salted caramel syrup in a medium bowl. Add the milk gradually, one tablespoon at a time, until the glaze is smooth and thick enough to coat the hand pies without running off completely.

Allow the fried hand pies to cool slightly, then dip the tops into the salted caramel glaze or drizzle it generously over each pie. Let the glaze set briefly before serving.