Preheat oven to 325°. Put coconut and butter in a medium bowl and stir until coconut is moist. Divide mixture evenly among tart pans and press into bottoms and up sides to form a thin shell. Put tart pans on a baking sheet and bake until shells are golden brown, rotating sheet halfway through, about 15 minutes. Let cool 5 minutes, then gently remove shells from tart pans and cool fully on a cooling rack.
In a medium saucepan over high heat, combine diced pineapple, pineapple juice, and star anise. Bring mixture to a boil, then reduce heat and simmer, uncovered, until pineapple is tender, about 5 minutes. Using a slotted spoon, transfer pineapple to a bowl and refrigerate, uncovered, until cool, about 10 minutes. Meanwhile, increase heat to high and boil remaining liquid until thick and syrupy and reduced to 1/2 cup, about 2 minutes. Remove star anise and discard; refrigerate syrup, uncovered, until cooled to room temperature, about 20 minutes.
In a medium bowl, stir mascarpone and pineapple syrup together. In a separate bowl, beat cream until stiff peaks form. Stir half the cream into mascarpone mixture to lighten it, then gently fold in remaining cream.
Divide cream filling evenly among tart shells, smoothing it with the back of a spoon or a small offset spatula. Spoon diced pineapple over cream filling, dividing evenly. Sprinkle the top of each tart with a pinch of crushed pink peppercorn. Serve immediately.
Note: Nutritional analysis is per tartlet.