Lovely : MayaHawke
The duo were seen in the Hamptons on July 21, where they appeared to be heading toward a local beach.
Hawke shares both Maya and her brother, Levon, 14, with Thurman.
There was a very small window of shooting time since those places like to stay open, keeping the dollars pouring in.
Hawke was married to his Gattaca costar for seven years before they split in 2005.