Heat the oven to 375°F (190°C). Peel, halve, and core the apples, cutting very large apples into quarters while keeping the pieces fairly large.

On a lightly floured surface, roll the puff pastry into a circle slightly larger than a 9- or 10-inch ovenproof skillet. Chill the pastry while preparing the caramel and apples.

Place the butter and sugar in the skillet over medium heat. Cook for 5–8 minutes, until the sugar dissolves and the mixture reaches a deep golden amber color, gently swirling the skillet instead of stirring aggressively.

Briefly remove the skillet from the heat and stir in the lemon juice, optional vanilla, and salt. Take care because the caramel may bubble vigorously.

Arrange the apples tightly in the skillet with the rounded sides down and the cut sides facing upward. Pack them closely because they shrink as they cook.

Return the skillet to medium heat and cook the apples in the caramel for 8–12 minutes. Continue until the apples begin to soften and the caramel thickens slightly.

Lay the chilled pastry over the apples and tuck the edges down inside the skillet. Cut two or three small vents in the pastry to let steam escape.

Bake for 30–35 minutes, until the pastry is deeply golden and puffed. Keep the skillet level when moving it so the hot caramel stays contained.

Let the tart rest for 5–10 minutes. Place a large serving plate over the skillet and carefully invert the tart in one decisive motion.

Rearrange any apples that remain in the skillet and spoon the extra caramel over the tart. Serve warm with crème fraîche, lightly whipped cream, or vanilla ice cream.