A cast-iron skillet works best when making this tart because it helps the fruit to caramelize, but any heavy-duty nonstick pan with an ovenproof handle will do.
Heat oven to 400 degrees. Core apples using a melon baller to retain the rounded shape of the apples. Rub apples with sliced lemons to prevent browning. Set aside.
Over medium heat in a 10-inch cast-iron skillet, heat sugar and 1/4 cup water. Cook until it turns a rich golden color; swirling occasionally, about 6 to 7 minutes. Meanwhile, roll puff pastry in one direction out into a 14-inch circle on a lightly floured surface, about 1/8 inch thick. Brush off any excess flour with a pastry brush.
Add butter to skillet. Once melted, add apples, rounded sides down, fitting as many into the skillet as possible. Remove from heat, and top apples with puff pastry. Fold edges under, and crimp slightly. Make two to three small cuts in puff pastry to allow the steam to escape. Place in oven for 10 minutes. Reduce heat to 375 degrees. and bake until pastry is golden brown, about 20 to 25 minutes.
Remove from oven, and let cool about 2 minutes. Loosen edges with a knife. Carefully place a plate over the skillet and flip, inverting the tarte tatin. Serve warm or at room temperature.