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.