For the tarte tatin: Trim the apple wedges so they each have a flat inner surface. Place the apples, 1/2 cup of the sugar and the lemon juice in a large bowl and toss to combine. Set aside for 20 to 30 minutes so the apples can release some excess liquid.
Preheat the oven to 375 degrees F.
Melt the butter in a 9-inch cast-iron skillet over medium heat. Add the remaining 1 cup sugar, the apple cider vinegar and vanilla and cook, stirring occasionally, until the mixture becomes a pale caramel shade, about 10 minutes. Drain the apples (discard the liquid) and arrange them flat-side up in the skillet, working in circles from the outside in, stacking and nestling as you go. (You want to have a taller mound in the center since the apples will shrink down as they cook.) Continue cooking until the sauce becomes a deep golden amber and the apples are tender when pierced, approximately 25 to 35 minutes, occasionally spooning the sauce over the apples while they cook.
Remove the puff pastry from the refrigerator. If necessary, roll the sheet on a lightly floured surface so it is at least 1 inch larger than the skillet. (I prefer to keep it a rectangular shape.) Place the pastry sheet over the cast-iron skillet and tuck the edges over the sides. Place the skillet on a rimmed baking sheet lined with parchment paper and transfer to the oven. Bake until the pastry is puffed and golden brown, 30 to 35 minutes. Remove from the oven and allow to cool for 10 minutes.
Place a large, heatproof serving plate upside-down over the skillet. Working quickly and carefully (obviously oven mitts are required), invert the skillet onto the plate. Rearrange any apple slices that may have moved around.
For the bourbon whipped cream: Whisk the heavy cream in a mixing bowl (or on the lowest setting in a stand mixer) for 30 seconds or so. Add the bourbon, confectioners' sugar and vanilla and continue to whisk until soft peaks just begin to form.
To serve, cut into wedges and serve with bourbon whipped cream.