Add brown sugar, butter, rum, vanilla and salt to an 8” cast iron skillet.
⅓ cup brown sugar, 1 tbsp. butter, ¼ cup dark rum, 1 tbsp. vanilla extract, ¼ tsp. salt
Cook over medium heat, stirring often, until the mixture starts to thicken and darken (about 8 minutes). Be careful not to burn it.
Lower heat to medium-low and add the peaches (cut side down) to the skillet. Use enough peach halves to fill up the bottom of the pan in one layer.
4-6 yellow peaches
Cook until slightly softened, about 5 minutes. Take the skillet off the heat.
Unroll the puff pastry sheet. Cut out a circle that is about 9” in diameter.
Place the puff pastry round on top of the peaches and tuck edges into the pan.
Bake in the preheated oven until the pastry is golden and puffed, 35–40 minutes.
Remove from oven and let cool on a wire rack for 30 minutes.
To invert the Tarte Tatin onto a serving dish: run a butter knife along the edge of the pastry to separate it from the skillet. With oven mitts, hold the serving dish on top of the skillet and carefully flip it all over. It should release with a couple of gentle shakes.
Dig in! It can be served with vanilla ice cream, if desired.