Rince the peaches and blot dry. Cut in half, discard pit, then cut into thick slices. Skewer two peach slices on each cinnamon stick with a mint leave between.
Combine butter, brown sugar, rum, cinnamon and salt in saucepan and bring to a boil over high heat. Let the glaze boil until thick and syrupy, about 5 minutes.
Prepare and preheat grill to high. Brush and oil the grate. Place the skewered peaches on the hot grate and grill until nicely browned. 3-4 minutes per side, basting with rum glaze. Remove from grill, spoon remaining glaze over peaches and serve.
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