Place a large zipper bag inside a bowl. Add the water, teriyaki sauce, garlic, lime margarita mixer, tequila, Worcestershire sauce, and ginger into the bag. Zip closed and shake gently to combine the marinade. Place the chicken into the bag, zip closed, cover and refrigerate for at least one hour.
To prepare the sauce – In a medium-size bowl combine the mayonnaise, yogurt, salsa, half-n-half, Cajun seasoning, parsley, and hot sauce. Whisk to until the sauce is combined. Cover and refrigerate until needed.
To prepare the Pico de Gallo – In a medium bowl combine the tomato, red onion, green onion, jalapeno pepper, and lime juice. Salt to taste and mix well. Cover and refrigerate until needed.
Spray the grill grates of your outdoor grill with a gluten-free high-heat grill spray. Pre-heat the grill to medium-high heat.
Remove the chicken breasts from the marinade, using about half of the sauce mixture divided into a separate bowl, brush each chicken breast on both sides to coat, then place onto the grill. Cook the chicken turning frequently and brushing with the sauce mixture on each flip. Grill until the internal temperature of the meatiest part of the breast reaches 165ºF. Remove from the grill to an oven-safe sheet when done. Spread remaining reserved sauce mixture over the breasts, sprinkle the grated cheese over the top and return briefly to the grill or broil in the oven just until the cheese is melted on top of the chicken.
Remove the chicken and top with Pico de Gallo, and garnish with reserved red and green onion to garnish for serving.