Heat olive oil in a large pot over medium heat. Add chopped onion and diced poblano, sauté until softened, about 5 minutes. Add minced garlic and cook 30 seconds until fragrant.
Season the chicken with salt, pepper, cumin, and smoked paprika. Sear in the pot 2–3 minutes per side until lightly browned.
Pour in chicken broth, bring to a simmer, cover, and cook until chicken is cooked through, about 10–12 minutes.
Remove the chicken, shred with two forks, and return shredded meat to the pot.
Stir in black beans and corn, then add half-and-half or cream. Simmer gently 5 minutes. Adjust seasoning and finish with lime juice.
Ladle into bowls and top with cheese, cilantro, avocado, and crushed tortilla chips as desired.
Notes
To make it lighter, substitute whole milk for the cream or stir in Greek yogurt off heat.
Use a rotisserie chicken to save time.
Freezes well for up to 3 months; thaw overnight before reheating.