Mix butter, garlic, parsley, dill, and lemon juice in a small bowl. Shape into a log, wrap in plastic, and freeze until firm.
Pound chicken breasts flat between two sheets of parchment until about 1/4 inch thick.
Place a piece of the frozen butter mixture in the center of each chicken breast. Roll tightly, tucking in the edges, and secure with toothpicks.
Set up three bowls: one with flour, one with beaten eggs, and one with breadcrumbs. Dredge each chicken roll in flour, dip in eggs, then coat with breadcrumbs.
Heat vegetable oil in a skillet over medium heat. Fry the chicken rolls until golden brown on all sides.
Transfer to a baking sheet and bake at 350°F (175°C) for 20 minutes until cooked through.
Remove toothpicks before serving and slice carefully to reveal the melted butter inside.