Make the marinade. Whisk together the olive oil, white wine vinegar, lemon juice, garlic, parsley, basil, oregano, salt, pepper, and mayo until smooth.
Marinade the chicken. Combine the chicken and the marinade in a freezer bag, seal the bag, and massage the marinade into the chicken. Refrigerate for 1 hour.
Prep. Preheat the oven to 375°F and grease a baking dish with butter.
Bake. Arrange the chicken in a single layer in the baking dish and pour the marinade over it. Bake for 15 minutes.
Make it cheesy. Sprinkle the parmesan and pecorino over the chicken and bake for 10 more minutes or until a meat thermometer inserted into the largest piece reads 165°F.
Broil. If you want your cheesy topping to brown up a bit more, set the oven to broil and broil the chicken for a minute or two.
Garnish, rest, and serve. Arrange the chicken on a serving platter and garnish with fresh parsley. Allow it to sit for 5 minutes or so before serving and/or slicing.