Marinate Chicken: Combine all Marinade ingredients in a bowl. Add chicken breasts, ensuring they are fully coated. Marinate for at least 30 minutes, or up to 4 hours in the refrigerator.
Preheat oven to 375°F (190°C). Lightly grease a baking dish.
Combine all Sauce ingredients in a bowl and mix well.
In a separate shallow dish, combine Panko bread crumbs and Parmigiano-Reggiano cheese.
Remove chicken from marinade and pat dry. Dip each chicken breast into the Sauce, then dredge in the Panko mixture, pressing to coat.
Place coated chicken breasts in the prepared baking dish.
Top each chicken breast with a slice of Monterey Jack cheese and a sprinkle of Gruyere cheese.
Bake for 25-30 minutes, or until chicken is cooked through and cheese is melted and bubbly.
