Combine all ingredients for compound butter. Using plastic wrap or wax paper, roll butter into a log, about 1 1/2 inches thick, and refrigerate for at least 1 hour. Chicken: Cut each breast into 2 halves. Remove the tip and first joint of each wing. With the back of a paring knife push the remaining skin down to the breast so that the wing bone is exposed. Detach the fillet strip, or "tender," from each half. Using a meat pounder gently flatten each breast and "tender" between sheets of plastic wrap to an even thickness of about 1/2-inch. Arrange breasts with the bone and smooth-side down on a work surface. Sprinkle breast pieces with salt and pepper and put a chilled slice of the compound butter in the center of each breast. Place a flattened tender over the butter. Enclose the butter with the chicken by folding bottom edge up over butter, tightly roll side edges jelly-roll style and secure with toothpicks. Arrange chicken, seam side down, in a shallow baking dish. Cover and chill for at least 4 hours.
Dip breasts in flour, then in egg, then in bread crumbs to coat evenly. Chill coated breasts for 1 hour. Preheat oven to 350 degrees.
In a large heavy skillet heat 1/2-inch of vegetable oil to 360 degrees F. or until a bread cube turns golden brown in about 30 to 60 seconds. Carefully lower two coated breasts into hot oil and fry until golden brown, about 3 to 4 minutes. Transfer browned chicken to paper towels to drain. Allow oil to reheat to proper temperature. Fry remaining two breasts. Place fried chicken in a shallow baking dish and bake, uncovered, for 15 minutes. Serve immediately.