Divide the butter between two 6-inch-diameter/15 cm, high-sided copper pans and spread it over the bottom of the pans. Sprinkle the sugar evenly over the butter in each pan. Using a small sharp knife, scrape the seeds from the vanilla beans into the sugar mixture. Reserve the vanilla beans.
Place the pans over medium heat and cook until the butter and sugar have melted, the sugar should start to turn a light golden color; once this has happened, remove from the heat. Place four apple halves, core-side facing up, onto the butter and sugar in each pan.
Place 2 cinnamon stick halves and 1 reserved vanilla bean between the apple halves in each pan to resemble a cross. Stretch the pastry over the apples and tuck the pastry down between the apples and the sides of the pans. Prick the pastry 4 to 5 times with a fork. Place the tarts in the refrigerator to chill the pastry for 1 hour. Preheat the oven to 375°F/180ºC.
Transfer the tarte tatin to the oven and bake for 20 to 25 minutes, or until the pastry is golden brown. Let rest at room temperature for 20 minutes to allow the flavors to meld.
To make the caramel sauce: Place the sugar in a medium saucepan over medium-high heat. Cook until the sugar dissolves, watching carefully once the sugar has dissolved since the sugar begins to change color and will burn quickly.
Allow the sugar to turn a golden brown. Remove the pan from the heat and add the brandy.
Allow to mix in thoroughly and return to heat. Add the cream and stir to blend. Once the cream is incorporated, remove from the heat and allow to cool slightly.
To serve tarte tatin: Place the pans with the tarts over high heat until the caramel in the bottom of the pans begins to bubble. Place a plate over each pan. In a swift movement, invert the tarte tatins onto the plates. Discard the vanilla bean and cinnamon stick. If the plate is a little messy, it may be easier to transfer the tarte tatins onto clean plates to serve. Drizzle the caramel sauce around tarte tatins. Serve each tarte tatin to two people.