Adjust oven rack to lower third position. Preheat oven to 375°F (190°C). Lightly grease a 9×13-inch baking dish (or two 8×8-inch dishes) with coconut oil or butter.

Spread ½ cup unsweetened shredded coconut on a small parchment-lined baking sheet. Toast in preheating oven for 4–6 minutes, stirring once, until golden and fragrant. Let cool completely.

In a large bowl, combine pineapple, cooled toasted coconut, granulated sugar, brown sugar, cornstarch, lime juice, vanilla, nutmeg, and salt. Gently fold with a silicone spatula until evenly coated. Let stand at room temperature for 15 minutes to macerate.

In a medium bowl, whisk flour, baking powder, baking soda, salt, and ¼ cup granulated sugar. Grate cold butter (or coconut oil) into mixture. Cut in with a pastry cutter or fingers until mixture resembles coarse meal with some pea-sized pieces. Stir in coconut milk and buttermilk just until shaggy dough forms—do not overmix. Fold in ⅓ cup toasted coconut.

Pour fruit mixture into prepared baking dish, spreading evenly. Drop batter by heaping ¼-cup portions across surface, leaving ½-inch gaps between mounds. Do not flatten.

Sprinkle generously with turbinado sugar and additional toasted coconut. Place dish on a parchment- or foil-lined half-sheet pan. Bake 45–55 minutes, rotating pan halfway, until topping is deep golden brown, puffed, and springy—and fruit is bubbling vigorously at edges.

Let cobbler rest untouched for at least 30 minutes (ideally 1–2 hours) before serving. Serve warm or at room temperature, topped with vanilla ice cream or coconut cream and extra toasted coconut.