Trim apple quarters to give each quarter a flat inner surface. To a large mixing bowl, add Apple quarters and toss with 1/2 cup sugar and the lemon juice. Set aside for half an hour.
Preheat oven to 375 degrees F.
To a 10-inch cast iron skillet set over medium-high heat, add 4 tablespoons butter. When the butter has melted and the foam has subsided, add 1/4 cup juice accumulated from the apples and the remaining cup of sugar. Lower the heat to medium-low and cook the mixture, stirring occasionally with a wooden spoon, until it turns a pale caramel color (10-12 minutes) – spoon some of the sauce on a white plate to check the caramel color.
Remove the skillet from the heat and carefully add the Apple quarters, arranging them flat side up in concentric circles. Cut remaining 2 tablespoons butter into small pieces and scatter them evenly over the apples.
Set Skillet back over medium-low heat and cook until caramel turns a deep amber color (18-20 minutes). While apples are cooking, occasionally spoon the caramel over the apples to allow them to cook evenly. Remove skillet from heat and set aside.
To prepare the pastry: place thawed puff pastry sheet on lightly floured surface. Using a rolling pin, roll the pastry to an even 1/4 inch thickness. Using a 10-inch cake pan as a template and a sharp knife, cuts a circle in the pastry, roll it around the rolling pin, then carefully unroll it over the apples in the skillet, tucking the edges between the apples and side of the skillet.
place the skillet on a rimmed baking sheet and transfer to the preheated oven. Bake until pastry rises and turns golden brown (30-35 minutes)
Remove the baking sheet from the oven and cool tart for 15 minutes.
Place a round serving plate upside-down over the skillet and invert the tart onto the plate, do it quickly & carefully. Cut the tarte tatin into wedges and serve immediately.