Sauté the base. Heat the olive oil in a large pot or Dutch oven over medium heat. Add the onion, jalapeño, and bell pepper. Cook until softened, about 6 minutes.
Bloom the spices. Add the garlic, chili powder, cumin, smoked paprika, and oregano. Stir constantly for 1 minute until fragrant — this toasts the spices and deepens their flavor.
Add tomatoes and broth. Pour in the fire-roasted tomatoes and chicken broth, scraping up any browned bits from the bottom of the pot.
Poach the chicken. Add the chicken breasts whole. Bring to a gentle simmer (not a rolling boil) and cook for 15 to 18 minutes, until the chicken is cooked through and shreds easily.
Shred. Remove the chicken to a cutting board and shred with two forks. Return it to the pot.
Finish the soup. Stir in the black beans, corn, and salt. Simmer 5 minutes more. Off the heat, stir in the lime juice and cilantro. Taste and adjust salt and lime — the soup should be bright and well seasoned.
Serve. Ladle into bowls and let everyone top their own: a handful of tortilla strips, avocado, cheese, a dollop of sour cream, and an extra squeeze of lime.