Preheat your oven to 400°F (200°C).

In a 10-inch cast-iron skillet or oven-safe pan over medium heat, combine the butter, maple syrup, brown sugar, cardamom, and vanilla extract.

Cook the mixture, stirring occasionally, until it bubbles and thickens slightly into a loose caramel sauce, about 3 to 5 minutes.

Remove the skillet from the heat. Arrange the pear halves in the caramel mixture, cut-side up, fitting them snugly together in a circle.

Return the skillet to medium heat and cook for another 5 to 8 minutes to let the pears soften slightly in the bubbling caramel.

Roll out the puff pastry on a lightly floured surface until it is slightly larger than the skillet.

Drape the pastry over the hot pears, carefully tucking the edges down inside the rim of the skillet to cup the fruit.

Cut three small slits in the center of the pastry to allow steam to escape.

Bake for 25 to 30 minutes, or until the pastry is puffed and deep golden brown.

Remove from the oven and let cool for exactly 10 minutes. Place a large serving platter over the skillet and carefully invert the tart onto the plate to serve warm.