Bette Midler received the female equivalent for her role in Hello, Dolly! and celebrated her second Tony win with a lengthy acceptance speech. And while Platt is a Tony rookie, he has a strong Broadway pedigree.
Other winners included August Wilson's "Jitney", which collected the Tony for best play revival.
As Zaks laughs his way through an interview, it is clear how much the 70-year-old actor-turned-director relishes his turn at the helm.
"Shut that crap off!" the veteran comedienne snapped at one point when the orchestra tried to play her off the stage.
The show, about a teenager who fabricates a former friendship with a classmate who killed himself, also saw Pitch Perfect star Ben Platt named best actor in a musical.
Yet other big names, among them Danny DeVito, Cate Blanchett and singer Josh Groban, left New York's Radio City Music Hall empty-handed.
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Pour ce parti anti-immigration et europhobe, c'est "une déception", a reconnu son vice-président Florian Philippot. Si Emmanuel Macron obtient la majorité absolue, "ce serait un exploit politique total".
Nixon was named best featured actress in a play for a revival of Lillian Hellman's "The Little Foxes", about a greedy southern family's underhanded business practices.
Sally Field, a nominee for "The Glass Menagerie", delivered a brief history of the service work done since the World War I era by "the women of the Wing" - the American Theater Wing, which administers the Tony Awards.
"Oslo" performer Michael Aronov won the Tony Award for best featured actor in a play, while Gavin Creel of "Hello, Dolly!" won best featured actor in a musical.
"Come From Away", the Canadian musical won just one: for best direction, by Christopher Ashley.
Spacey even had fun with rumors about his sexual orientation while singing the Andrew Lloyd Webber song "As If We Never Said Goodbye" from "Sunset Boulevard". Appearing as his "House of Cards" character Frank Underwood, he wisecracked that he'd better leave, "before Bette Midler thanks anyone else". Dressed as Glenn Close, who stars in that revival, Spacey sang, "I'm coming out." and then added, "Of makeup". Spacey talked about the show's accountants and said: "You guys do not have to worry about them tonight, at all", he said.
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