Combine coconut milk, evaporated milk and cinnamon sticks, in large saucepan set over medium-low heat; bring to a simmer.
Meanwhile, beat egg yolks with sweetened condensed milk electric mixer until thick and pale yellow. Gradually whisk in half the hot milk mixture into egg yolk mixture. Pour into saucepan. Cook, stirring constantly with wooden spoon or spatula, for 5 to 8 minutes or until mixture thickens and coats back of spoon.
Remove from heat. Strain through fine-meshed sieve and remove cinnamon sticks. Cool completely.
Stir in rum or aguardiente and vanilla. Cover and chill in refrigerator for at least 6 to 8 hours and up to 3 to 5 days.
Pour into small glasses to serve and garnish with toasted coconut and cinnamon.