Combine the butter, parsley, tarragon, 1 teaspoon salt, and 1/4 teaspoon black pepper in the bowl of a stand mixer. Beat until well combined. Place the mixture on plastic wrap or wax paper and roll into small log, and freeze while preparing the chicken.
Set the chicken breasts, one at a time, on a large piece of plastic wrap, lightly spritz the top of the chicken with water, fold the plastic wrap over the chicken, and spritz the top of the plastic wrap with water. Pound to no less than 1/8-inch thick. Season each piece of chicken with salt and pepper.
Lay a breast on a new piece of plastic wrap and place one quarter of the compound butter and 1 tablespoon of the panko in the center of each breast. Using the plastic wrap to assist, fold in the ends of the breast and roll into a tight log, completely enclosing butter. Repeat with each breast. Refrigerate the breasts for 2 hours or up to overnight.
Place egg and water mixture and the remaining panko into separate shallow dishes. Heat 1/2 inch of vegetable oil in a 12-inch sauté pan to 375ºF over medium-high heat.
Dip each breast in the egg mixture and then roll in the breadcrumbs. Gently place each breast in oil, seam-side down, and cook until golden brown, 4 to 5 minutes per side, or until the internal temperature reaches 165ºF. Remove to a wire rack set in a half sheet pan to drain and rest for 5 to 10 minutes before serving.