Mix 1 cup water and white rice in a bowl and stir to combine. Cover and let steep in the fridge for at least 4 hours and up to overnight.
Strain the rice mixture through a fine mesh sieve into a separate bowl, leaving rice behind. You can either discard the rice or save it for another use.
Put rice water, additional ½ cup water, sugar, and cinnamon stick in a medium saucepan and bring to a boil.
Once boiling, lower heat and let mixture simmer for about 10 minutes, until syrup has thickened (note that it will continue to thicken as it cools). Remove from heat and stir in vanilla extract.
Pour syrup into a mason jar or bowl and place in fridge to cool completely before using. Keep cinnamon stick in syrup for at least 4 hours for full cinnamon flavor.