Soak rice in water in a large pot, stirring occasionally, about 30 minutes.
Drain the water into a medium, 2-quart pitcher.
Blend rice and cinnamon in a blender or food processor, until pasty and rice is well-broken. Transfer rice paste in the water. Let stand 2 to 3 hours, stirring occasionally.
Strain the rice water through a fine sieve or cheesecloth into a large, 3-quart pitcher. Add sweetened condensed milk, whole milk, vanilla extract, cinnamon sticks; stir briefly. Refrigerate 8 hours or overnight.
Stir in coconut rum and spiced rum until flavors are well-mixed.