Peel, core, and quarter all apples from top to bottom.
Melt the butter in an 8 to 10 inch heavy-bottomed skillet over medium heat. Arrange the apple quarters round-side down, crowding them to cover the entire bottom of the pan.
Pour the sugar evenly over the apples. Add the kosher salt and cinnamon if using.
Cook over medium heat, swirling the pan as needed, until the caramel reaches a medium amber. Rotate the pan for even color. Be careful not to over or under-develop the caramel.
If using, add the apple brandy to the pan and carefully ignite using a gas burner, kitchen torch, or lighter. Allow the flame to burn out completely before proceeding.
Remove from heat and cool to room temperature for 15-20 minutes.
Preheat the oven to 375°F conventional.
Cut the puff pastry into a circle just slightly larger than the pan. Lay it over the apples, tucking in any overhanging edges.
Bake for 20 to 30 minutes, until the puff pastry is fully puffed and deep golden brown.
Cool in the pan for 10 minutes, then carefully invert onto a large plate or serving dish. Slice while still warm for the cleanest cuts.
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Notes
Apples: Choose firm, moderately tart varieties such as Honeycrisp, Gala, Crispin, Crimson Crisp, or Ginger Gold. Avoid soft varieties like Macintosh, which turn mealy under prolonged heat. Flipping: Rest the tart in the pan for the full 10 minutes before inverting. Flipping too soon allows the caramel to run before it has had a chance to firm slightly. Storage: Store cooled tarte tatin in an airtight container or tightly wrapped in the refrigerator for up to 3 days. The puff pastry will soften over time, but the flavor holds. Best served the day it is made.