Prepare the chicken breasts by placing them between two sheets of plastic wrap and gently pound to an even thickness of about 1/2 inch.
Make the garlic butter by mixing softened butter, minced garlic, chopped parsley, chopped dill, lemon juice, salt, and black pepper in a bowl until well combined.
Assemble the chicken by placing half of the garlic butter mixture in the center of each pounded chicken breast, folding the sides over the butter, and rolling it up tightly.
Bread the chicken by dredging it in flour, dipping it in beaten eggs, and rolling it in breadcrumbs until well coated.
Heat vegetable oil in a skillet over medium heat and fry the chicken rolls for about 4-5 minutes on each side until golden brown. Then transfer to a baking dish and bake in a preheated oven at 375°F for 15-20 minutes until cooked through.
Let the chicken rest for a few minutes before serving to allow the juices to redistribute.