In a large bowl or sealable plastic bag combine water, teriyaki sauce, juice of ½ lime, garlic, tequila, liquid smoke, salt and ginger. Place the chicken breasts into the bowl and mix well into the marinade. Cover and refrigerate for at least 1 hour.
To prepare the sauce—in a medium bowl combine mayonnaise, sour cream, salsa, milk, Cajun seasoning, parsley and hot sauce. Cover and refrigerate until needed.
To prepare the Pico de Gallo—in a medium bowl combine tomatoes, red onion and jalapeno pepper. Add 2 tablespoons of lime juice, season with salt and mix well. Cover and refrigerate until needed.
Slightly grease the grates of an outdoor grill. Heat the grill to medium-high heat. Remove the chicken breasts from the marinade and place onto the grill. Cook the chicken turning frequently until fully cooked (the internal temperature has reached 165 degrees F.). During the last few minutes of cooking, brush each chicken breast with the sauce and then sprinkle with cheese. Cook until the cheese has melted.
Remove the chicken from the grill and top with Pico de Gallo for serving.