Anne Burrell’s girlfriend Koren Grieveson is moving here to be with the Food Network star. We’re told Grieveson, also a top chef, just left Avec in Chicago, where she worked since 2003. A friend said, “Anne is meeting Koren in Chicago to help with the move. They will drive Koren’s car to New York.” The spiky-haired blond star of “Chef Wanted With Anne Burrell” just celebrated her 43rd birthday at McKittrick Hotel’s Gallow Green with pals including Alex Guarnaschelli and Claire Robinson.