On the stovetop, preheat a grill pan (that has an accompanying press) on medium high heat.
In a medium size bowl, add brown sugar, melted butter (reserve 1 Tbsp. of butter to brush on grill), peach liqueur (or orange juice) and cinnamon. Stir until sugar dissolves.
Add peach halves and toss to coat.
Using a silicone pastry brush, brush 1 Tbsp. of the reserved butter on both the grill pan and press. Place peach halves on grill, flesh side up, and cover with press.
Grill for 5 minutes. Remove peaches from grill using nylon tongs or spatula. Serve in individual bowls and garnish with cinnamon sticks.
Optional: Serve with fresh whipped cream or vanilla ice cream.