In a medium saucepan, combine the water with the lime zest and 1 cinnamon stick and simmer over moderate heat for 15 minutes. Strain into a large heatproof bowl and refrigerate until chilled, at least 1 hour.

Whisk in the sugar, coconut milk, evaporated milk, light rum, vanilla and egg yolks, if using; whisk until the sugar is dissolved. Refrigerate the drink until chilled, about 2 hours. Stir well and ladle into small chilled martini or cordial glasses; garnish with the remaining cinnamon sticks.