In a large mixing bowl, whisk together the flour and salt. Add the cold, cubed butter and cut it into the flour using a pastry blender, two knives, or your fingertips, until the mixture resembles coarse meal with some pea-sized pieces of butter remaining.

Gradually add the ice water, 1 tablespoon at a time, mixing lightly with your hands until the dough just comes together. Be careful not to overmix. Form the dough into a flat disc, wrap tightly in plastic wrap, and refrigerate for at least 30 minutes, or up to 2 days.

Peel, core, and quarter the apples. If using larger apples, you might need to cut them into sixths or eighths. Try to keep the pieces roughly similar in size for even cooking. Set aside.

Preheat your oven to 375°F (190°C).

In your 9-10 inch oven-safe skillet, melt the softened butter over medium heat. Add the granulated sugar and stir until the sugar dissolves and the mixture starts to bubble. Continue to cook, stirring occasionally, until the caramel turns a rich amber color. This will take about 8-12 minutes. Remove from heat immediately to prevent burning.

Carefully arrange the apple quarters tightly in a concentric pattern over the caramel in the skillet. Start from the outside edge and work your way in. Try to pack them in as snugly as possible, standing them upright if necessary, as they will shrink during cooking. If using, drizzle vanilla extract over the apples.

Return the skillet to medium heat and cook the apples, gently turning them occasionally, for about 15-20 minutes, or until they have softened slightly and released some of their juices into the caramel. This also helps to further develop the caramel flavor.

On a lightly floured surface, roll out the chilled pastry disc into a 10-11 inch circle, slightly larger than your skillet. Carefully place the pastry over the apples in the skillet, tucking the edges down around the apples and into the sides of the pan.

Cut a few small slits in the top of the pastry to allow steam to escape during baking.

Bake for 25-30 minutes, or until the crust is deeply golden brown and puffed, and the apples are tender when poked through the pastry vents.

Carefully remove the skillet from the oven. Let it cool for 5-10 minutes on a wire rack. This cooling period is crucial, as it allows the caramel to set slightly and prevents it from spilling everywhere when inverted.

Place a large serving plate (one with a slight rim is best to catch any caramel) upside down over the skillet. Using oven mitts for protection, firmly grasp both the skillet and the plate, then quickly and confidently flip them over. Lift the skillet carefully; if any apples stick, gently dislodge them with a spatula and place them back onto the tart.

Serve the Tarte Tatin warm, ideally with a dollop of crème fraîche, vanilla ice cream, or whipped cream.