Throw the remaining marinade ingredients into the bag along with the chicken halves. Store in the refrigerator for at least 12 hours. Turn the bag periodically.
Before broiling, remove the garlic and cumin seeds from the skin of the chicken.
Arrange the chicken halves - end to end on the grate, directly under the broiler. Broil for about 15 minutes. (Baste every 5 minutes by squeezing lime juice on the chicken.) Re-arrange the halves after the first 10 minutes. Turn the halves over and broil for another 15-20 minutes, basting and re-arranging as before. Turn the chicken halves for the final time. Season with salt and chili powder. Layer slices of Longhorn cheese over both halves. Season again with chili powder. Broil until bubbly. Remove from the oven, and allow to rest for 8 minutes.