Brown the chicken in half the oil, season lightly, then remove.
Add the remaining oil, then cook the onion, poblano, and potatoes until the onion is softened and the potatoes are just beginning to soften.
Stir in the garlic for 30 seconds, then add the cumin, chili powder, smoked paprika, and oregano. Stir 30 seconds.
Add the broth and 1½ cups of the corn. Simmer until the potatoes are tender.
Blend the remaining ½ cup corn with a little broth until smooth, then stir it into the soup.
Return the chicken to the pot and simmer until cooked through.
Stir in the half-and-half and heat gently. Taste and adjust seasoning.
Serve with Monterey Jack and optional tortilla strips, cilantro, and lime.