Place the rice, water, lemon peel, cinnamon sticks, and salt in a large saucepan. Bring to a boil, reduce heat to low, and cook uncovered until the rice is soft, about 25 minutes or so. Remove the lemon peel and cinnamon sticks.
Add the coconut milk, whole milk, shredded coconut, and sugar to the rice and cook over very low heat, uncovered, stirring frequently until thick, about 30 min to one hour. At this point the rice should be very sticky with most of the milk absorbed. TIP: Keeping the heat low will reduce the number of stirrings.
Now, gently stir in the heavy cream and continue cooking over low heat, just long enough to warm the rice completely.