Preheat a stovetop grill over high heat. Cook corn for 15 minutes, turning every 5 minutes. Transfer corn to a cutting board. Slice kernels from the cob once cool enough to handle.
Heat oil in a large soup pot over medium-high heat. Add onion and jalapeno. and cook for 4 minutes. Add garlic and cook for 1 minute more. Reduce heat to medium.
Add broth, diced green chilies, chili powder, cumin, lime juice, and corn kernels. Cook until heated through, about 5 minutes.
Place sour cream in a large bowl. Ladle 1 scoop of soup broth over sour cream. Whisk until combined. Repeat this step. It is important to temper the sour cream so that it does not curdle once it goes into the soup.
Stir the tempered sour cream into the soup. Add pepper Jack cheese and chicken. Cook until cheese is melted, about 10 minutes, stirring frequently.
Ladle soup into bowls and garnish with cilantro. cotija, and lime wedges.
