Break the bread into pieces. In a large bowl, soak the bread in the half-and-half until the liquid is absorbed, about 2 minutes. Mix in the chicken, egg, salt (if any), pepper, and the 3 tablespoons room-temperature butter. Put in the freezer for about 10 minutes to firm up.
Remove the chicken mixture from the freezer; it will still be very soft. Form the mixture into four oval cutlets. Mix the parm and bread crumbs in a small bowl and coat the cutlets with the bread crumb mixture.
In a large, nonstick frying pan, heat the remaining 2 tablespoons butter and the oil over moderate heat. Cook the cutlets until golden brown and just done, 4 to 5 minutes per side.