Reduce heat and simmer, stirring occasionally to prevent cheese from sticking to bottom of pot, until broth is flavorful and reduced by half, about 2 hours.
To prevent the chicken from sticking to the bottom of the pot, hold each piece with tongs or a spider in the hot oil just above the bottom for about a minute to allow the bottom skin of the chicken to seal.
In a medium saucepan, combine the milk, coconut milk, rice and cinnamon stick over medium - high heat, stirring every few minutes to prevent the rice from sticking to the bottom of the pot.