Combine the sugar and cinnamon stick with 1 cup water in a small saucepan and bring to a simmer, stirring until the sugar is dissolved. Simmer until slightly reduced, about 3 minutes. Remove from the heat, stir in the vanilla and let stand for 10 minutes. The syrup can be made up to 3 days ahead and kept refrigerated; reheat until warm before serving.
Meanwhile, prepare a grill for medium heat, making sure the grates are very clean. Brush the cut-sides of the peaches and one side of the pound cake slices with the butter.
Grill the peaches, cut-side down, until well marked, about 5 minutes. Transfer to a cutting board and cut into wedges. Grill the pound cake slices, buttered-side-down, until lightly marked, about 2 minutes.
Serve each slice of pound cake with the wedges of half a peach; drizzle with syrup and top with a dollop of whipped cream.