In a small bowl, mix together the softened butter, minced garlic, chopped parsley, dill (if using), lemon zest, salt, and pepper. Shape the herb butter into a log, wrap it in plastic wrap, and refrigerate until firm, about 30 minutes.
Place each chicken breast between two sheets of plastic wrap. Using a meat mallet or rolling pin, gently pound the chicken to an even thickness of about ¼ inch (6 mm). Season both sides with salt and pepper.
Cut the chilled herb butter into four equal pieces. Place one piece of herb butter in the center of each flattened chicken breast. Fold the sides over the butter, then roll the chicken tightly to enclose the butter completely. Wrap the chicken tightly in plastic wrap and refrigerate for 30 minutes to firm up.
Prepare three shallow dishes: one with flour, one with the beaten eggs, and one with breadcrumbs. Dredge each chicken roll in flour, shaking off excess. Dip into the beaten eggs, allowing excess to drip off. Coat with breadcrumbs, pressing gently to adhere.
Preheat the oven to 350°F (175°C).
In a large skillet, heat about 1 inch (2.5 cm) of oil over medium-high heat. Fry the chicken rolls until golden brown on all sides, about 2-3 minutes per side. Transfer to a paper towel-lined plate to drain excess oil.
Place the fried chicken rolls on a baking sheet. Bake in the preheated oven for 15-20 minutes, or until the internal temperature reaches 165°F (74°C).
Allow the chicken to rest for 5 minutes before serving. Serve hot, garnished with fresh parsley if desired. Pair with mashed potatoes, steamed vegetables, or a side salad.