Well, originally, Chile was an Aztec food, not 'Mexican'. If you prefer the term 'soup' to 'gravy' OK with me. Mole (chile, chocolate, peanutbutter sauce) is an Aztec dish. Served over chicken or turkey with rice.

Real green chile (made with New Mexico/Anaheim green chiles) goes well with pork and chicken, but beef prefers the red. I've got loads of great recipes if anybody is interested. Lived in New Mexico a few years and green chile seems to be one of the things that any one who has ever lived there takes away with them. Green Chile Rellenos are super!