Melt the butter in a saucepan until it bubbles. Add the sugar and stir with a wooden spoon until smooth. Add the vanilla bean and cinnamon stick and stir to combine, pressing on the vanilla bean to bring out the vanilla seeds inside. Continue simmering until caramel in color, about 5 minutes. The mixture might not look smooth and blended at this point.
Stir in the brandy, lemon juice and orange zest and continue cooking until smooth, about 5 minutes more. Heat the oven to 375 degrees F and arrange the ramekins on a cookie sheet. Pour 2 tablespoons of caramel into the bottom of each ramekin. Arrange 1 peach half in each ramekin. Pour a little more caramel over the peaches and bake until the caramel is bubbling and the peaches are almost tender, 15 to 20 minutes. (The recipe can be made up to this point, cooled to room temperature, and refrigerated overnight. Bring to room temperature before proceeding.)
Meanwhile, use a cutter or sharp knife, cut the puff pastry into 12 rounds, slightly larger than the diameter of your ramekins (they will shrink during the baking). Remove the ramekins from the oven and turn it up to 425 degrees F. Top each ramekin with a round of puff pastry and bake until the pastry is golden brown, about 10 minutes more. Let cool at least 1 hour or up to 8 hours. When ready to serve, re-warm the tarts slightly to loosen the caramel by placing them in a shallow pan of very hot water, or on a flame tamer on your stovetop, set over medium heat. Unmold upside down onto small plates and serve, with the peaches up and the pastry on the bottom.
Recipe courtesy of Gale Gand's Just A Bite by Gale Gand and Julia Moskin: Clarkson N. Potter Publishers, 2001