Preheat oven to 350 degrees Fahrenheit. Bake puff pastry rounds until golden brown, about 10 to 15 minutes. When golden, remove and let cool.
Sautée pineapple with coconut cream, brown sugar, pineapple juice, coconut, and cinnamon sticks. When pineapple caramelizes, remove from heat and carefully add the dark rum. Return to the heat and carefully flambé. Cook until the alcohol cooks off and the flames die out. Remove from heat.
To assemble, in large serving bowl or cup, place puff pastry round first in the center. Scoop out ice cream and place on the round. Serve warm pineapple-rum sauce over ice cream. Garnish with extra toasted coconut or with half cinnamon stick.
