Preheat the oven to 425°F.
In an 8-inch cast-iron skillet, melt the butter over medium heat, then sprinkle the sugar over it. Cook until the sugar melts and caramelizes, stirring occasionally to make sure it doesn’t burn, 6 to 8 minutes. Stir in the lemon juice. Drop the cardamom pods, cinnamon sticks, ginger, and star anise in the middle. Arrange the apples in a circular pattern, cut-sides up, in the caramel.
Cook slowly over low heat until the apples start to absorb the caramel and the juices are bubbling around them, 25 to 30 minutes, depending on the size of the apples. Turn the apples so they cook evenly.
Once the apples are coloring nicely and the caramel is bubbling, remove from the heat and drape the puff pastry over the top, tucking it around the apples along the edge of the skillet.
Bake for 20 to 25 minutes or until the pastry is golden brown. Remove from the oven and invert the skillet onto a serving platter that is large enough to catch the excess caramel.