PREPARATION:

1. Preheat the oven to 400 °F/200 °C.   In a heavy medium saucepan, cook the sugar over medium heat without stirring for 5 minutes, or until melted.  Stir in the lime juice. Add the butter and maple syrup, and stir until well blended and the mixture thickens slightly, about 8 minutes. Stir in the coconut. Remove from the heat.
2. Slice each apple into 4 horizontal rings and remove the core. Working with 1 apple at a time and using about 1/2 cup of coconut mixture for each, place the bottom apple piece on a 12-inch square sheet of foil. Spoon some of the coconut mixture over the apple piece. Repeat layering the apple pieces and coconut mixture.
3. Carefully wrap the apple stacks in the foil and place them on a baking sheet. Bake for 10 minutes, or until the apples are crisp-tender and juices have formed in the foil packages.
4. Peel back the foil from each apple to expose the apples but cradle the juices. Continue baking 15 minutes longer, or until the coconut mixture is golden brown and the apples are very tender.
5. Transfer each package to a large plate. Tear away the foil, allowing the sauce to pool onto the plates. Gently slide the apples atop the sauce and serve.