1 (14- to 16-ounce package) frozen puff pastry, thawed according to the package directions
Flour, for dusting the work surface
2 tablespoons packed dark brown sugar
1/2 teaspoon ground ginger
1/8 teaspoon fine salt
2 cups fresh, small-dice pineapple
Heat the oven to 375°F and arrange a rack in the middle. Generously coat a 12-well muffin pan with the melted butter using a pastry brush. Evenly sprinkle 1 teaspoon of the coconut inside each well; set the pan aside.
Roll out the puff pastry on a lightly floured surface to 1/8-inch thickness. (Some brands may not require any rolling.) Prick it all over with a fork, then cut 12 circles using a round cookie cutter the width of the top of the muffin wells (about 2 3/4 inches). Place the puff pastry circles on a baking sheet lined with parchment or waxed paper and refrigerate until needed.
Place the brown sugar, ginger, and salt in a medium bowl, break up any lumps with a spatula or wooden spoon, and stir to combine. Add the pineapple and stir to coat.
Evenly divide the pineapple mixture among the muffin wells. Place a puff pastry circle on top of the pineapple mixture and gently tuck the sides into the well to seal around the pineapple.
Bake until the pastry is puffed and dark golden brown, about 30 to 35 minutes.
Remove the pan from the oven and let cool on a wire rack for 5 minutes. While the tartlets are still warm, run a knife around each to loosen it from the well. Place a baking sheet over the muffin pan and carefully invert the muffin pan onto the baking sheet. Gently tap the bottom of each well and then remove the muffin pan. (If some pieces of pineapple and coconut remain inside the wells, just scrape them out and place them on top of the tartlets.) Serve warm.