Position rack in center of oven and preheat at 400 degrees F. Arrange apples, round sides down, in deep heavy 10-inch skillet (skillet will be crowded). Cook sugar and butter in another skillet over medium-high heat until sugar is dark amber color, stirring constantly, about 10 minutes. Immediately pour sugar over apples. Cover skillet and cook apples over medium heat until just tender, about 15 minutes. Drain off 1/3 cup pan juices and reserve. Cover skillet and cook apples 10 minutes longer. Remove skillet from heat and uncover.
Lay 1 puff pastry sheet out on lightly floured surface. Sprinkle pastry evenly with cinnamon. Place second pastry sheet atop first. Trim corners to make rough circle. Pierce pastry several times with fork. Place pastry over apples in skillet, tucking pastry around apples at edges of skillet. Bake until pastry is puffed and golden, about 20 minutes.
Remove skillet from oven. Using oven mitts as aid, place large platter over pastry and invert skillet onto platter. Tap skillet firmly in several places to loosen apples. Carefully remove skillet. Boil reserved apple pan juices in small saucepan over high heat until reduced to 1/4 cup, about one minute. Brush over hot apples. Cut tart into wedges and serve hot or warm with whipped cream or vanilla ice cream.
