Combine butter, parsley, tarragon, 1 teaspoon salt, and 1/4 teaspoon black pepper in the bowl of a stand mixer. Place mixture on plastic wrap or waxed paper and roll into small log; place in freezer.
Place chicken breasts, 1 at a time, between 2 pieces of plastic wrap. Squirt chicken lightly with water and squirt the top of the plastic wrap as well. Pound to no less than 1/8-inch thickness. Season each piece of chicken with salt and pepper.
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 in 1 pie pan and 2 cups bread crumbs in a different pie pan.
Heat 1/2-inch of vegetable oil in a 12-inch saute pan over medium-high heat until oil reaches 375 degrees F.
Dip each breast in the egg mixture and then roll in the bread crumbs. Gently place each breast in oil, sealed-side down, and cook until golden brown, approximately 4 to 5 minutes on each side, until the internal temperature reaches 165 degrees F. Remove to a cooling rack set in sheet pan and allow to drain for 5 to 10 minutes before serving.