“Chicago” star Christie Brinkley took on the role of doctor to help a woman who fainted at the Tonys, with Paul Shaffer playing nurse. Backstage at the Beacon — where stars congregated to snap up swag before the show — Brinkley’s “gift-suite escort” keeled. Sources said David Letterman‘s bandleader immediately grabbed orange juice and fruit for a sugar boost, while Brinkley fanned and reassured the dazed young victim until paramedics arrived. Brinkley was later rewarded for her good deed: She won a diamond watch worth $35,000 in the gift suite.