Slice each chicken breast into two thin cutlets. Place a cutlet on a cutting board with a plastic wrap. Cover it with another layer of plastic wrap. Pound the chicken gently with a flat side of a meat mallet (or rolling pin) to an even thickness of about 0.5 cm (¼ inch). Set aside and repeat with the process with the rest of the meat.
Take out the butter from the freezer and place it on a cutting boar. Cut the log half and then cut each half lengthwise, so you have 4 long pieces. Set it aside on a separate plate.
Place one pounded chicken cutlet on a cutting board. Season it with about ¼ teaspoon of salt and some pepper. Place one chunk of butter in the middle of the chicken cutlet, closer to the thinnest end. Fold the sides of the chicken breast over the herb butter, then roll it up tightly, making sure to tuck in any protruding chicken to ensure the herb butter is fully sealed inside. Repeat with the rest of the cutlets.
Arrange rolled chicken breasts on a baking pan or a plate lined with parchment paper and place them in the freezer for at least 30 minutes. Chilling them will help to seal the meat better.
In a three separate shallow dished, add flour, whisked eggs and bread crumbs. Remove the chicken from the freezer. Take a cutlet and coat it in flour, shaking off any excess. Dip it into beaten eggs, ensuring a thorough coating, and transfer it to a dish with breadcrumbs. Use your hands to coat the chicken with breadcrumbs, gently pressing them to adhere to the meat. Repeat the process by dipping it back into the egg mixture and then into the breadcrumbs. You can choose to do a single coating or double coating for your cutlets; double-coated ones tend to be more uniform and crispier. Repeat with the remaining 3 cutlets.
After coating the cutlets, place them in the refrigerator for at least 15 minutes. Meanwhile, preheat the oven to 400°F and line a baking pan with parchment paper.
In a medium pot, heat canola or vegetable oil until it reaches 350°F. Carefully, add 2-3 chicken cutlets into the pot and fry them for approximately 4-5 minutes turning them half way, until they turn golden brown. Use a slotted spoon to remove the cutlets and place them on the prepared baking dish. Repeat this process for the remaining chicken.
Bake the cutlets for about 15-17 minutes in a preheated oven or until they are fully cooked. Look for tiny bubbles forming on top of the bread coating as an indication of doneness. To double-check, you can also use a meat thermometer by inserting it into the side of the cutlets, making sure it reaches 165°F. Avoid inserting the thermometer into the very center to prevent the melted butter from leaking out.
Once the chicken is fully cooked, remove it from the oven and serve it hot alongside your favorite side dish.