Heat the oil in a large pot or Dutch oven over medium heat. Add the onion, bell pepper, jalapeno, garlic, salt and black pepper and cook, stirring, until the veggies are tender, about 7 minutes.
Add the smoked paprika, oregano, and cumin and stir to combine, toasting the spices for 1 to 2 minutes, being careful not to burn.
Pour in the broth, enchilada sauce, and green chiles. Bring to a rapid simmer.
Add the chicken and the rice to the soup and stir to combine. Reduce the heat so that the soup is gently simmering, about medium-low heat. Cover and cook until the rice is tender and the chicken is cooked through, about 20 minutes.
When ready, using tongs transfer the chicken to a sheet pan and using two forks, shred the chicken.
Add it back into the soup along with the can of black beans (drained and rinsed), lime juice, and cilantro. Stir to combine and let simmer gently for about 5 more minutes, to allow the black beans to heat through and the flavors to meld.
To serve, ladle into bowls and garnish as desired.