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In a saucepan, combine the rice and 1 1/2 cups milk together and cook over medium heat, stirring frequently, until the milk is mostly absorbed, about 10 minutes.
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Meanwhile, in a small bowl, whisk together the remaining 1/2 cup milk, the egg, and the white sugar.
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Pour the mixture into the rice on the stove and stir, letting it cook, for a few minutes, until it has thickened. Add the brown sugar and cook another minute or two, stirring to keep from burning on the bottom.
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Remove from heat and stir in butter, vanilla, and cinnamon. Let sit for about 10 minutes to thicken up and serve warm, or let cool completely in the fridge.