all-purpose flour, for work surface
1 sheet frozen puff pastry, thawed
1 pound frozen pineapple chunks, thawed
1/4 cup sugar
1 tablespoon lemon juice
2 and 1/2 teaspoons powdered unflavored gelatin (one 1/4-ounce packet)
3/4 cup cold heavy cream
1 mango, peeled, pitted, and thinly sliced
4 ounces unsweetened coconut flakes (2 tablespoons), toasted
Preheat oven to 400 degrees. On a floured surface, roll puff pastry into a 10-by-14-inch rectangle. Trim uneven edges. Place pastry on a baking sheet. With a sharp knife, lightly score pastry 1 inch from edges, creating a border; with a fork, prick dough all over inside border. Bake until golden, 12 to 15 minutes. Let cool completely.
Meanwhile, in a medium saucepan, cook pineapple, sugar, and lemon juice over medium-high until pineapple begins to break down, 10 to 12 minutes. Transfer to a blender and puree until smooth. Pour through a fine-mesh sieve back into saucepan, discarding solids, and sprinkle with gelatin. Let stand until gelatin softens, 5 minutes. Bring to a boil over medium-high, stirring, the pour into a medium bowl set in a bowl of ice water. Stir until cool and thickened, 4 minutes.
Whisk cream until soft peaks form. Stir pineapple mixture until smooth; fold into whipped cream until combined. Spread on pastry and top with mango and coconut.