The board said the terrace would disrupt neighbors with loud noise.
Attendees of a packed public forum called for new zoning regulations and a moratorium on skyscrapers.
The Wednesday night panel will address infrastructure, landmarks and shadows cast by the giant towers.
James Lomma, who was cleared in a fatal 2008 East Side crane collapse, owned the rig, officials said.
The Department of Buildings has issued more than 300 after-hours permits for One57 in the past year.
Chefs at Le Parker Meridien hotel built a gingerbread model of the crane knocked down by Hurricane Sandy.
The area around ONE57 has been closed since a crane there partially collapsed during Hurricane Sandy.
The crane has been dangling over West 57th Street since Hurricane Sandy began battering the city Monday.
Condos inside ONE57, which will be the tallest residential tower in New York, start at $6.375 million.
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