This tart bakes best in a cast-iron skillet.
On a lightly floured work surface, roll out an eighth of the quick puff pastry dough to about 1/8-inch thickness. Using a sharp paring knife, cut dough into a circle about 7 1/2 inches in diameter. Transfer to a parchment-lined baking sheet, and place in refrigerator to chill, about 15 minutes.
Preheat the oven to 425 degrees. Place the chilled pastry dough in a heavy cast-iron skillet that measures 6 1/2 inches in diameter at the bottom. Peel, core, and slice apple in half lengthwise. Using a mandoline or very sharp knife, slice apple halves widthwise into twenty-five 1/8-inch-thick slices. Arrange the slices in a neat fanned pattern, overlapping the slices and keeping them 1/2 inch in from the edge of the pastry dough; fill in the center with smaller or broken apple slices as you create the fanned-out circle.
Sprinkle 2 tablespoons sugar over the apple slices, and dot with 1 tablespoon butter cut into very small pieces. Place skillet in the oven, and bake until the pastry has puffed up along the edges of the pan and is golden brown, about 30 minutes.
Remove the skillet from the oven. Using a spatula, remove the tart from the skillet, and transfer to a plate; set aside. Add the remaining tablespoon butter to the skillet, and place over medium heat. Add the remaining 2 tablespoons sugar, and cook until the sugar dissolves and forms a light caramel, about 5 minutes. Measure brandy into a glass, then pour into the caramel; cook out the alcohol, 2 to 3 minutes.
Using a second plate, invert the tart so it is upside down. Return the tart to the skillet, apple side down, and cook 4 to 5 minutes, until the caramel bubbles up over the tart, and starts to look a little thick. Remove the skillet from the heat, and carefully invert the tart onto a plate large enough to catch the hot caramel as it drips out of the skillet. Serve immediately.