Preheat the oven to 400 degrees F.
Mix the sliced apples together with 1/2 cup of the sugar and the lemon juice in a large bowl and set aside. The lemon juice will give flavor as well as keep your apple wedges from oxidizing. Melt the butter in a 12-inch nonstick ovenproof skillet over medium-high heat and sprinkle the remaining 1/2 cup sugar evenly over the melting butter.
Place the apples in the skillet, flat-side down, starting from the outside and working in, slightly overlapping one another. Fill in the center with the remaining apples. Return the skillet to medium-low heat on the stovetop and cook until the caramel darkens in color evenly, 20 to 25 minutes. Keeping your heat on medium-low will ensure that your caramel doesn't burn and cooks evenly.
Remove the skillet from the heat. Lay the puff pastry over the skillet and simply fold the corners in to fit into the pan.
Bake until the crust has puffed and is golden brown, 20 minutes. Let cool, making sure that it is cool enough to handle. Loosen up the edges using a butter knife. Place a plate or platter larger than the skillet right on top. Carefully (but with gusto!) flip the tart upside-down onto the plate. If the apples stick to the skillet, carefully remove them and put them back where they belong on the tart.
Serve the warm tart with gelato or whipped cream.