Preheat the oven to 350 degrees F. Bake the puff pastry rounds until golden brown, about 10 to 15 minutes. When golden, remove and let cool.
Saute pineapple with coconut cream, brown sugar, pineapple juice, coconut, and cinnamon sticks. When pineapple caramelizes, remove from the heat and carefully add the dark rum. Return to the heat and carefully flambe. 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 round. Serve warm pineapple-rum sauce over ice cream. Garnish with some extra toasted coconut or with half cinnamon stick.