Heat oven to 375° F. Generously grease six of the cups in a standard 12-cup muffin tin with butter. Set aside.
On a generously floured surface, roll the pastry dough into an approximately 9 x 13-inch rectangle. Using a 4" biscuit cutter, a small bowl, or the lid to a take-out container, cut 6 large circles from the dough. Transfer circles to a parchment lined baking sheet and chill for 10 minutes. Roll any scraps into a ball, wrap it with plastic film, and chill.
In a large bowl, combine peaches, lemon juice, rum, brown sugar, nutmeg, ginger, and corn starch.
Remove chilled dough rounds from refrigerator and, working quickly to prevent them from getting sticky, press each one into a greased muffin cup, pressing gently against the bottoms and sides to prevent air bubbles. Using a slotted spoon, spoon the peach mixture into the prepared muffin tins, filling each one generously to the top of the tin in a rounded mound. Chill for 10 minutes.
Roll out chilled scraps on a floured surface. To make lattice tops, slice the scraps into ½-inch strips and overlap them across the filled pie cups, crimping and pressing the ends into the edges of the bottom crust to seal them. Brush dough with beaten egg and sprinkle with turbinado sugar.
Bake pies for 30 minutes until the crusts are deeply golden and filling is bubbling. Let cool in the pan for 15 minutes. Run a butter knife gently around the pie's edges to loosen them, then transfer them to a rack to cool completely.