Preheat oven to 350 degrees. Test each of the three eyes at stem end of coconut to see which is the softest. Then use an ice pick (or a screwdriver) and a hammer to pierce two of the eyes. Strain water through a fine sieve into a bowl, and reserve.
Place coconut on a rimmed baking sheet; bake for 15 minutes. Remove the coconut, carefully wrap it with a kitchen towel, and place it on a solid, stable surface. Hit the area around the eyes of the coconut forcefully with a hammer or kitchen mallet. Remove the hard shell. Use a vegetable peeler to remove the brown skin from the back of the coconut meat. Rinse flesh under cold running water. Finely grate coconut using a box grater. Transfer to a large glass measuring cup. Add reserved coconut water and rum; set aside.
Working in batches, in the jar of a blender, combine coconut milk, cream of coconut, condensed milk, evaporated milk, and coconut rum mixture; blend until combined. Pour into a pitcher, and add cinnamon sticks. Refrigerate, covered, for up to 2 days. Just before serving, stir to combine. Serve over ice.