Put 2 cans black beans, diced tomatoes with green chile peppers, 1 cup chicken broth, cumin, salt, black pepper, and cayenne pepper into a blender and blend until smooth; pour into a saucepan and add remaining chicken broth and 1 can black beans. Place saucepan over medium heat, bring mixture to a simmer, and cook for 15 minutes.
Put 1/2 cup hot rice into each of 6 bowls. Ladle soup over rice and top with 1 tablespoon sour cream.
