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Prepare the rice and keep warm.
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Meanwhile, grill (or broil) the chicken breasts over medium heat, 6 to 8 minutes per side, until tender and no longer pink. Shred the chicken with two forks. Season with salt and pepper.
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To make the sauce, put all the sauce ingredients in the blender and blend until smooth. Heat the sauce in a saucepan or the microwave.
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Heat the black beans in a saucepan or the microwave.
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To assemble burritos: Divide the chicken among the 6 tortillas. Add a scoop of rice and a scoop of beans to each one. Fold up the bottom third of one tortilla over the filling, fold in the sides, and then fold down the top. Repeat for the remanning tortillas.
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Preheat the broiler to low and adjust the oven rack to about 6 inches from the heat. Place the filled burritos seam side down in a broiler pan, or oblong metal pan, and pour the sauce over all. Sprinkle with the cheese.
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Place the pan under the broiler. Watch closely and broil until the sauce is bubbly, just a few minutes.
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Serve with chips and salsa.