Note: You can make compound butter ahead of time and keep in the fridge for up to a month or in a freezer for up to 3 months. Make sure to wrap it tightly in plastic wrap and place it into a zip-lock bag for storing.
Take butter out of the refrigerator and let it sit on the counter for 30-45 minutes. Make sure it is softened first so it's easy to mix with other ingredients.
Mix butter with minced herbs, garlic, lemon zest, salt, and pepper. Use a fork to mix and make sure that all ingredients are evenly incorporated with butter.
Spread butter in a log-like shape on top of a sheet of plastic wrap and wrap it up tight into a log.
Place into the freezer for about 15 minutes while preparing the chicken.
To butterfly chicken breasts: place the chicken breast onto the cutting board place your hand on top, make sure to slightly bend your fingers up and not curl or angle them down. Slice through the center of the chicken all the way to the other side but not cutting it in half.
Open the chicken like a book (or butterfly wings...hence the name) and place it flat into a zip-lock bag.
Use the flat side of the meat tenderizer to pound the chicken, working from the center outward to the edges. Make sure the meat is between 1/4 and 1/8 of an inch but not thinner or it will top. Also, make sure not to hit too hard so it doesn't break the meat.
Take the chicken breast out carefully and set aside. Repeat with remaining chicken breasts.
Season the chicken with salt and pepper on both sides.
Wrap the chicken and butter:
Take butter out of the freezer and out of the wrap. Slice the butter log into 8 even slices. You will use 2 per chicken breast.
Place chicken breast onto a slice of plastic wrap. Make sure there is enough of plastic wrap to wrap the chicken tightly in a couple of layers.
Place two slices of butter, one next to another, onto one side of the chicken breast and start rolling the butter and chicken together, tucking the sides in as you roll. Once you have a tight chicken pocket, roll it in plastic wrap tightly.
Repeat with the other three chicken breasts and butter.
Place rolled chicken into the refrigerator for at least 2 hours or overnight.
NOTE: you can prepare the chicken the night before and then, batter and cook it the next day. Just don't exceed 24 hours before cooking the chicken.
Batter the chicken:
Preheat the oven to 350° and add oil to the large skillet for frying. Preheat the oil to 350°. Line a rimmed baking sheet with aluminum foil and place a wire rack inside.
Prepare three shallow and wide bowls for battering the chicken. In one bowl, whisk together eggs with some salt and pepper. In another bowl, whisk together flour with some salt, pepper, and garlic powder. In a third bowl, mix together Panko bread crumbs with dried herbs, and some salt and pepper. (Make sure to only use a little salt and pepper in each mixture so it's not overly salted!)
Take chicken out of the refrigerator and carefully unroll it from the plastic wrap. Work carefully to batter the chicken but not disturb the tight roll. Dredge chicken rolls in flour first, then egg mixture and the Panko crumbs. Press it in with your hands and repeat, dredging it in flour first, then in egg, and then in Panko mixture. Press with your hands and set aside. Repeat with remaining chicken rolls.
Make sure the oil is heated to 350° and carefully add battered chicken using long metal tongues. (Be very careful not to press too hard and not scrape any batter off the chicken.)
Fry chicken for about 3-4 minutes on each side, until deep golden. Take it out of the oil and place onto the prepared baking pan, on top of the wire rack.
Bake for 20-24 minutes, depending on how large were your chicken breasts.
(Since you can't pierce the chicken to check internal temperature, you have to judge by the size of your chicken breasts. If you used chicken breasts that are smaller than 6 oz. each, you may need to use less compound butter inside and bake them for 15-20 minutes.)
*Note that for smaller chicken breasts, you should use a little less compound butter. Don't overstuff the chicken or it might break or leak.