4 large ripe freestone peaches
8 (3-inch) cinnamon sticks
8 fresh mint leaves
4 tablespoons unsalted butter
1/4 cup firmly packed brown sugar
1/4 cup dark rum
1/2 teaspoon ground cinnamon
Peach or vanilla ice cream, for serving
Rinse the peaches and blot them dry with paper towels. Cut each peach in half and discard the pit. Then, cut each peach into quarters. Using a pointed chopstick or metal skewer, make a starter hole in the center of each peach quarter, working from the pit side to the skin side. Skewer 2 peach quarters on each cinnamon stick, placing a mint leaf between the 2 quarters.
Combine the butter, brown sugar, rum, cinnamon, and salt in a saucepan and bring to a boil over high heat. Let the glaze boil until thick and syrupy, about 5 minutes.
Prepare and preheat the grill to high. Brush and oil the grate. Next, place the skewered peaches on the hot grate and grill until nicely browned, 3 to 4 minutes per side, basting with the rum and butter glaze. Spoon any remaining glaze over the grilled peaches and serve at once. Peach or vanilla ice cream makes a great accompaniment.