The classic French caramelized tart is made easier using a muffin pan.
Preheat oven to 400 degrees. Place puff pastry circles in refrigerator. In a small saucepan, combine brown sugar, rum, and pinch of salt; bring to a boil over medium-high heat. Boil 1 minute, then add butter and stir until completely melted. In a jumbo muffin pan, divide sugar mixture among the cups, about 4 teaspoons each. Divide mango among cups and top with pastry. Bake until pastry is golden brown, puffed, and juices are bubbling, 20 to 25 minutes, rotating pan halfway through.
Let cool on a wire rack, 15 minutes. Run a small knife around edges of pastries to loosen. Place a rimmed baking sheet over pan and quickly invert to release tartlets. Serve warm.