Place a chicken breast on a cutting board and, using a sharp knife, slice horizontally almost through to create a pocket or open like a book so the breast lies evenly thin; 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.
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 or roll the chicken over the butter and press the edges firmly to seal the filling; trim or pin if needed to keep sealed.
Set up three shallow dishes: one with the flour mixed with 1 teaspoon salt, one with the beaten eggs, and one with the panko breadcrumbs.
Dredge each sealed chicken breast in the flour (shake off excess), dip in the beaten eggs, then press evenly into the panko to coat completely.
Heat oil in a large skillet over medium heat until it reaches about 350°F (hot but not smoking) or is hot enough that a breadcrumb sizzles on contact.
Fry the breaded chicken breasts in the skillet, 4 to 5 minutes per side, until golden brown and cooked through; work in batches if needed to avoid crowding.
Transfer the cooked chicken to paper towels to drain briefly, sprinkle with chopped fresh parsley, and serve immediately.