Heat oil in a large soup pot and cook onion for 5 minutes. Add garlic, tomato paste, chili powder and cumin and stir for 1 minute.
Sprinkle in masa harina and stir for 30 seconds, then gradually whisk in broth until smooth.
Add chicken and salt and simmer gently for 15 to 18 minutes, stirring occasionally, until the chicken reaches 165°F and the broth lightly thickens.
Lower the heat and stir in milk. Remove from the heat and add cheddar by handfuls, stirring until melted.
Stir in lime juice and serve without returning the cheese-enriched soup to a hard boil.