Bring water to a boil in a medium saucepan; add cinnamon sticks and lime peel.

Reduce heat and simmer covered for 5 minutes.

Add rice and return to a boil.

Simmer, covered, for 20 minutes, or until all the liquid is absorbed and the rice is tender.

Stir in the milk, sugar, and salt.

Simmer over medium-low heat, stirring frequently, for 40 minutes or until most of the liquid is absorbed.

Remove from heat. Discard cinnamon and lime peel.

Preheat broiler.

In a medium bowl, combine egg yolks and vanilla.

Spoon a few tablespoons of the hot rice mixture into the egg yolk mixture.

Stir the egg yolk mixture and 1 1/4 cups of shredded coconut back into the rice mixture.

Transfer the rice mixture to a 2-quart square baking dish.

Top with the remaining 1/2 cup of coconut and the butter pieces; sprinkle with ground cinnamon.

Broil for 3-4 minutes or until coconut is golden brown.

Serve warm.