In a wide skillet, melt the butter over medium-low heat. Add in the sugar, cinnamon, nutmeg, vanilla extract and salt and whisk to combine. Add the apples to the caramel mixture and gently stir together, coating the apples in the caramel. Cook, stirring occasionally, until the apples are slightly tender, 5 to 8 minutes depending on the size of your apples. Note: If the mixture begins to get too thick, add some water to loosen it up.
Transfer the apples to a 10-inch nonstick ovenproof skillet and arrange them cut-side up, making sure to evenly cover the bottom of the skillet. Drizzle half the caramel sauce over the apples, then return the caramel to the heat and cook until thick and glossy, another 5 minutes. Set aside.
Roughly measure and cut the puff pastry sheet to approximately the size of the top of the skillet. Place the puff pastry on top of the apples and gently tuck the sides in ever so slightly. Cut the remaining scraps of puff pastry and evenly spread across the top (this will give the tarte tatin some volume).
Bake until the pastry is puffed up and golden brown, about 25 minutes. Let cool for 10 minutes, invert a plate on top and quickly (but carefully) flip onto the plate. Serve with vanilla ice cream or whipped cream and the remaining caramel sauce.