Position a rack in the center of the oven and heat the oven to 400°F (205°C). Use a 10-inch oven-safe cast-iron or stainless-steel skillet with a heatproof handle. Lightly flour a work surface if needed and roll the thawed puff pastry into an approximately 10-inch circle. Place it on a plate or baking sheet and refrigerate while you prepare the apples.

Arrange the apple quarters snugly in concentric circles in the cold skillet, rounded sides down. Set aside.

In a small saucepan, combine the sugar and water. Cook over medium heat without stirring until the sugar dissolves and the syrup turns a deep amber color, 6 to 9 minutes. Swirl the pan gently as needed for even coloring. Remove from the heat as soon as it is amber; caramel can darken quickly.

Carefully add the butter, lemon juice, salt, and vanilla to the caramel. It will bubble vigorously. Stir with a heatproof spoon until smooth, then immediately pour the caramel evenly over the apples in the skillet.

Set the skillet over medium heat. Cook until the caramel is bubbling and the apples begin to soften, about 15 minutes. Use tongs to reposition any apples that shift, keeping them in a snug layer. Remove the skillet from the heat.

Drape the chilled puff pastry over the apples. Quickly tuck the edge down around the apples inside the skillet, taking care not to touch the hot caramel. Cut 3 small steam vents in the pastry.

Bake until the pastry is deeply golden and crisp, 22 to 25 minutes. Let the tart stand in the skillet for 10 minutes; this brief cooling time helps the caramel settle while it remains fluid enough to release.

Wear oven mitts and place a large rimmed serving plate over the skillet. In one confident motion, carefully invert the skillet and plate together. Lift off the skillet, spooning any apples or caramel left behind onto the tart. Serve warm, ideally with vanilla ice cream or whipped cream.