In a small bowl, mix the softened butter with minced garlic, chopped parsley, chopped dill (if using), lemon juice, salt, and black pepper until well combined.
Shape the herb butter into a small log and wrap it in plastic wrap. Freeze until firm, about 30 minutes.
Place each chicken breast between two pieces of plastic wrap. Using a meat mallet or rolling pin, gently pound the chicken breasts to an even thickness, about 1/4 inch (0.6 cm) thick.
Season the chicken breasts with salt and black pepper.
Cut the frozen herb butter log into four equal pieces.
Place one piece of herb butter in the center of each chicken breast.
Fold the sides of the chicken breast over the butter, then roll up the chicken tightly, enclosing the butter completely. Secure with toothpicks if needed.
Place the flour in one shallow dish, the beaten eggs in another, and the breadcrumbs in a third dish.
Dredge each stuffed chicken breast in the flour, shaking off any excess.
Dip the chicken in the beaten eggs, allowing any excess to drip off.
Coat the chicken in the breadcrumbs, pressing gently to adhere. Repeat the egg and breadcrumb coating for a thicker crust if desired.
In a large skillet, heat about 1/2 inch (1.25 cm) of vegetable oil over medium-high heat until hot.
Carefully add the breaded chicken breasts to the skillet, seam side down.
Fry the chicken until golden brown on all sides, about 3-4 minutes per side.
Transfer the fried chicken to a baking sheet.
Preheat your oven to 375째F (190째C).
Bake the chicken breasts in the preheated oven for about 20 minutes, or until the chicken is cooked through and the internal temperature reaches 165째F (74째C).
Remove the toothpicks from the chicken breasts before serving.
Enjoy your meal!