Heat the oven to 400°F. Line 6 cups of a standard 12-cup muffin tin with circles of parchment paper.

Using a sharp knife or mandoline, very thinly slice the apples around the core into half-moon slices 1 to 1½ millimeters thick. Soak them in just-boiled water for 3 to 5 minutes, until they become soft and bendable. Don’t worry about them oxidizing—they’ll brown anyway in the caramel.

Roll out the pastry to 2 millimeters thick. Cut lengthwise into 6 strips. (Store-bought puff pastry usually comes folded into thirds, so you can use those creases as a guide.)

Arrange one-sixth of the apple slices on the top half of each strip, then fold the pastry in half and press down, leaving the rounded, unpeeled edges of the apple slices exposed at the top.

Roll up one apple-filled pastry strip to form a rose. Cut the excess dough off if necessary and pinch the end of the dough strip to seal. Repeat with the remaining strips.

In a small pot over medium heat, gently stir the sugar with a splash of water (just enough to make the sugar look like wet sand), then use a wet brush to make sure there are no sugar granules on the side of the pan. Cook undisturbed for about 10 minutes, until the sugar caramelizes to a deep amber (you can tilt the pan if needed for even cooking, but avoid stirring). Mix in the vanilla and cinnamon.

Pour 2 to 3 soup spoons of caramel into each muffin well. Arrange the apple roses upside down in each well.

Bake for 25 to 30 minutes, until the puff pastry is cooked through—it should feel firm to the touch, with a light golden color.

Allow the roses to rest for 5 minutes before serving.