Melt the butter in a large saucepan, soup pot, or Dutch oven over medium heat.
Add the chopped onion and poblano pepper along with half of the seasoning blend. Cook for 5–6 minutes, stirring occasionally, until the vegetables soften and develop a little color.
Add the shredded chicken, drained black beans, frozen corn, chicken broth, and remaining seasoning. Stir everything together.
Increase the heat to medium-high and bring the soup to a gentle simmer. Reduce the heat as necessary and simmer uncovered for 15 minutes.
Stir in the heavy cream and shredded cheese. Continue stirring gently until the cheese has completely melted into the broth.
Bring the Chicken Poblano and Black Bean Soup back to a gentle simmer and cook uncovered for another 15 minutes. This gives the flavors time to develop and allows the broth to thicken slightly.
Stir in the fresh lime juice. Taste and adjust the salt and black pepper if necessary.
Divide the hot soup among bowls and garnish with finely chopped cilantro before serving.