Rinse the peaches and blot them dry with paper towels. Cut each peach in half and discard the pit.

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. Let cool. Place peaches and glaze in a plastic bag for at least one hour. It’s better overnight.

Prepare and preheat the grill to high. Brush and oil the grate. Next, place the 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.