Combine butter, parsley, tarragon, 1 teaspoon salt, and 1/4 teaspoon black pepper in the bowl of a stand mixer. Beat until well combined. Place mixture on plastic wrap or wax paper and roll into small log; place in freezer.
Lay a breast on a new piece of plastic wrap and place one quarter of the compound butter and 1 tablespoon of the breadcrumbs in the center of each breast. Using the plastic wrap to assist, fold in the ends of the breast and roll into a log, completely enclosing butter. Roll very tightly. Repeat with each breast. Refrigerate the breasts for 2 hours or up to overnight.
Lay 1 chicken breast on a new piece of plastic wrap and place 1/4 of the compound butter and 1 tablespoon bread crumbs in the center of each breast. Using the plastic wrap to assist, fold in ends of breast and roll breast into a log, completely enclosing the butter; roll very tightly. Repeat with each breast. Place chicken in refrigerator for 2 hours, or up to overnight.
Place egg and water mixture and the remaining breadcrumbs 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, seal-side down, and cook until golden brown, 4 to 5 minutes per side, or until the internal temperature reaches 165ºF. Remove to cooling rack set in a half-sheet pan to drain and rest for 5 to 10 minutes before serving.