The two pals chowing down on Chinese backstage at the Marquis Theater in New York are Dixie Carter and Delta Burke, who is still in costume from her role as a down-on-her-luck actress who runs a sleazy hotel in the Broadway musical “Thoroughly Modern Millie.” The actresses appeared in two TV series together: the short-lived “Filthy Rich” in 1982 and the famously successful “Designing Women” from 1986 until Ms. Burke departed her role as Suzanne Sugarbaker after five seasons following her high-profile battle with the producers.