Place a chicken breast on a cutting board and, using a sharp knife, cut horizontally through the thickest part almost to the opposite edge to create a pocket; open like a book to make an even, thin piece. Repeat with remaining breasts.
In a small bowl, combine the softened butter, 1 1/2 tablespoons chopped parsley, minced garlic, and lemon-pepper seasoning until smooth and well blended.
Divide the herb butter into four portions (about 1 1/2 tablespoons each) and place one portion in the center of each opened chicken breast. Fold the chicken over and press the edges firmly to seal the filling inside.
Set up three shallow dishes: mix the flour with 1 teaspoon salt in the first, beat the eggs in the second, and place the panko in the third.
Dredge each stuffed chicken breast in the seasoned flour, shaking off excess, dip into the beaten eggs, then press evenly into the panko breadcrumbs to coat both sides.
Heat oil in a large skillet over medium heat until it reaches about 350°F (or is hot but not smoking). Add the breaded chicken and fry 4 to 5 minutes per side, or until golden brown and cooked through.
Remove the chicken from the skillet and drain briefly on paper towels. Sprinkle with chopped fresh parsley and serve immediately.
