The play for which Patty and Cathy get parts is William Shakespeare's "Antony and Cleopatra," one of the bard's historical plays first performed on stage in London, England, circa 1607.
Although this is the only appearance Alan Mowbray makes in the series and the only time he appears with Patty Duke in a production, he was a very prolific and sought-after character actor in a career that spanned four decades of work in the movies and on TV.
For her theatrical ambitions and accomplishments, Patty's father jokingly likens her to Helen Hayes, who was a renown stage actress in the early and mid-20th Century, as well as a recipient of the best actress award, i.e. the Oscar, from the Academy of Motion Picture Arts and Sciences in 1932 for her titular role in The Sin of Madelon Claudet (1931).
Patty's father teases her for her colloquialisms by suggesting her mode of speaking does not match that of Eleonora Duse, who was world-renown for her stage performances in the late 19th and early 20th centuries.
While he did put his face out front in a few of his productions, such as the driver who takes Cary Grant to the airport in The Bachelor and the Bobby-Soxer (1947), Dore Schary mostly kept behind the camera where he produced, co-wrote, and even directed scores of movies. Here he is represented by a very good look-alike, i.e. Walter Kattwinkel.