Position the oven rack in the top third of the oven and preheat to 425 degrees F. Spray an 8 by 3-inch round cake pan with nonstick spray.
Combine the sugar and 6 tablespoons water in a heavy medium saucepan. Stir over medium heat until the sugar dissolves. You will not stir the sugar after this point. Increase the heat and bring to a boil. Do not stir. Cook until the syrup is a deep amber color, 8 to 10 minutes. Remove the pan from the heat. Carefully add the butter (the caramel will bubble vigorously). Stir to blend, then add the flaked sea salt. Pour the caramel into the prepared cake pan. Let cool 5 minutes.
Peel, quarter and core the apples. Cut into 1/2-inch thick slices. You don't want to cut the slices too thin because they will fall apart or break down when cooked - the 1/2-inch slices will hold their shape. Fill a large bowl with water and add the lemon juice. Place the apple slices in the acidulated water and set aside. This will keep the apples white.
Place the cake pan over a stovetop burner, but don't turn the burner on. Lay 2 apple slices in the center of the pan on top of the caramel. Arrange the remaining apple slices in a circle, overlapping one another so that the end result looks similar to a rose. While layering, occasionally press down on the apples to make sure they all fit into the pan. If the caramel starts to set, turn the heat on very low to help loosen it up.
With an oven mitt (the pan may be hot), pop the pan into the oven. Bake until the apples are tender and the caramel bubbles thickly, about 1 1/2 hours.
Meanwhile, unfold the pastry sheet on a lightly floured work surface. Using a clean 9-inch round cake pan bottom as an aid, trace and cut out a 9-inch round from the pastry.
Remove the pan from the oven. Top the apples with the pastry round, pressing down and tucking in the edges. Pierce or dock the pastry all over with a fork. Return the pan to the oven and bake until the pastry is golden, about 25 minutes. Remove from the oven and cool the tart completely in the pan.
Invert a platter or cake stand over the cake pan. Hold the platter and the pan and invert. The tart will fall out onto the platter. If the tart does not cooperate, run the edge with a knife and place over medium heat until slightly warmed, then invert the pan onto the platter. Cut into wedges and serve with vanilla ice cream, if desired.