Heat 2 TBSP of olive oil in a large Dutch oven or stockpot over medium-high heat. Add the onion and poblano, and sauté for about 5 minutes, or until the onions start to sweat, stirring occasionally. Stir in the garlic and sauté until the garlic is aromatic—about a minute.
Add the chicken broth, tomatoes and juice, black beans, shredded chicken, lime juice, chili powder, cumin, salt, pepper, smoked paprika, and cayenne pepper. Stir well to combine and bring the mixture to a boil. Let it boil gently for about 5 to 7 minutes.
Finally, add the parsley or cilantro and simmer for another minute. Give the soup a taste test and add more salt or seasonings to taste. The flavor and saltiness will vary based on the chicken broth and personal preference. Make any adjustments you feel necessary with salt and other spices.
Spoon the soup into bowls and garnish with tortilla strips. You can also top with cubed avocado, cheese, and chopped fresh herbs. Some folks even add a dollop of sour cream—got nuts.