Bring a medium saucepan of salted water to a boil. Add the cinnamon sticks and cho-chos. Boil until tender, about 40 to 50 minutes. Drain and cool cho-chos until cool enough to handle. Peel the cho-chos, remove and discard the seeds, and roughly chop.Place in a food processor and pulse until mashed and smooth. Place the pulp in a fine-meshed strainer and using a rubber spatula, drain off any excess moisture from the mashed cho-chos. (Need about 1 cup of the mashed, drained puree.)
Pour oil into a heavy pan to a depth of 2 inches. Preheat oil to 360 degrees F.
Combine the flour, allspice, cinnamon, baking powder, salt, and sugar in a medium bowl; stir to combine well. Whisk together the cho-cho puree, eggs, vanilla extract, and melted butter in a separate bowl. Add the cho-cho mixture to the flour mixture, stirring just until combined.
Using a tablespoon scoop or small ice cream scoop, fry the fritters in batches in the preheated oil, until golden brown, about 4 to 5 minutes. Remove from the oil with a slotted spoon and drain on paper towels. Garnish with confectioners' sugar and serve immediately, drizzled with Coconut Rum Sauce.
Yield: about 2 1/2 dozen