In a saucepan over medium-high heat, bring the milk to a boil. Add in the rice and millet. Reduce the heat to medium-low and simmer until the rice and millet is fully-cooked (about 20-30 minutes).
Preheat the oven to 350°F. Place thin pats of butter on the bottom of a ceramic baking dish.
Remove the rice from heat.
Add in vanilla extract, cinnamon, and sugar. Stir in the pumpkin puree.
Spread the mixture evenly into the baking dish.
Bake the mixture for an hour, or until the sides are golden brown.