Heat the olive oil in a large stockpot or Dutch oven over medium heat.
Add the chopped poblano peppers and onions. Cook for 5 to 6 minutes, stirring occasionally, until the vegetables soften.
Stir in the minced garlic and cook for 30 to 60 seconds until fragrant.
Add the cubed chicken breasts, chili powder, ground cumin, kosher salt, black pepper, and cayenne pepper to the pot.
Cook for 4 to 5 minutes, stirring frequently, until the chicken browns on the outside.
Pour in the chicken broth, enchilada sauce, salsa verde, rinsed black beans, and drained corn. Stir well to combine.
Bring the soup to a gentle simmer. Cook for 10 to 15 minutes, or until the chicken reaches an internal temperature of 165°F (74°C) on a meat thermometer.
Remove the pot from the heat. Stir in the minced cilantro and lime juice.
Serve warm with optional toppings like sour cream, diced avocado, shredded cheese, and tortilla chips.