Marinate: Whisk together water, teriyaki, lime juice, garlic, tequila, liquid smoke, salt, and ginger. Add chicken, cover, and refrigerate at least 2 hours (up to overnight).
Make the sauce: Whisk mayo, sour cream, salsa, milk, Cajun spice, lime juice, parsley, hot sauce, and dill until smooth. Cover and chill.
Cook to 165°F: Grill 5–7 min/side, air-fry at 370°F for 10–12 min, pan-sear 4–6 min/side, or bake at 400°F for 20–25 min.
Sauce & melt: Brush chicken with sauce, top with Monterey Jack, and melt (air fryer 300°F for 3–5 min, lidded skillet, or broiler).
Serve: Plate over tortilla strips, drizzle with extra sauce, and finish with pico, cilantro, and lime. Serve with Mexican rice.