Place chicken between sheets of waxed paper and pound with a mallet until very thin, being careful not to tear meat.
In a food processor or blender, combine basil, cheese and garlic. Mix until a thick paste is formed. Add butter and blend. Shape into a 3-inch square. Chill until firm. Slice chilled square into 6 sticks, 1/2-inch wide and 3-inches long.
Lay out the flattened chicken breasts. Place a stick of pesto butter to one side of each piece of meat (preferably to the larger side). Fold the remaining side over butter, then fold down the two ends. Press meat together on all sides to securely enclose butter.
Using 3 separate pie plates, place seasoned flour in one, beaten egg in another and bread crumbs in the third.
Dredge each chicken breast in flour and shake off excess. Dip into egg, then roll in bread crumbs. Firmly pat bread crumbs onto chicken.
Refrigerate chicken rolls for at least 1 hour to allow crumbs to adhere. At this point chicken rolls may be frozen, but thaw completely before cooking.
In a large saute pan, heat butter and olive oil. Saute chicken on medium to medium-high heat for about 15 minutes, turning chicken until golden brown on all sides and meat is white all the way through.