In a large bowl, whisk together the first 8 ingredients (water, Teriyaki sauce, lime, garlic, Tequila, liquid smoke, salt, and ground ginger), coat chicken in mixture, and marinade for at least 2 hours or overnight.
In another large bowl whisk together the next 9 ingredients(mayonnaise, sour cream, spicy salsa, milk, Cajun spice, parsley, hot sauce, dill weed, and cumin), cover, and chill until needed.
Grill the marinated chicken breasts for 3-5 minutes per side, or until cooked through. Discard leftover marinade.
Brush grilled chicken with reserved dressing, sprinkle with cheese, and broil until cheese has melted.