Roast the poblano peppers. Turn a gas burner to high and place the poblanos directly on the grates. Using heat-proof tongs, turn the peppers occasionally until charred on all sides. Transfer to a bowl, cover with plastic wrap, and let steam for about 20 minutes. Once cool enough to handle, use a paper towel to rub off the charred skin. Remove the stems and seeds, then chop the peppers into small pieces.
Melt the butter. In a large soup pot or Dutch oven, melt the butter over medium heat.
Cook the onion. Add the onion, season with salt and pepper, and cook for about 5 minutes, stirring occasionally, until softened.
Add the seasonings. Stir in the roasted poblanos, garlic, oregano, cumin, and smoked paprika. Cook for about 3 minutes, stirring occasionally, until fragrant.
Make the creamy broth. Add the softened cream cheese and chicken broth. Cook for about 10 minutes, stirring occasionally, until the cream cheese has completely melted into the broth.
Blend until smooth. Use an immersion blender to blend the soup until smooth. Alternatively, carefully transfer the soup to a blender in batches and blend until smooth.
Add the chicken, beans, and corn. Stir the beans, corn, and shredded chicken into the blended broth. Simmer for 10 minutes, stirring occasionally.
Finish the soup. Turn off the heat. Stir in the lime juice and shredded cheddar cheese until the cheese is just melted.
Season and serve. Taste and add additional salt and pepper as needed. Serve warm with your favorite toppings.