Grant Davis, Carson City’s young, up-and-coming musician, has made a giant step up.Davis, 16, recently signed with Manhattan Records and began recording with the company Monday in Los Angeles.“It was a long wait for the contract to come through,” said Grant’s mom, Angie, “But now our manager says we’re going to be super-busy.”Manhattan made an offer in October, she said. Negotiations bounced back and forth through attorneys, slowed by the pounding the recording giant’s New York headquarters took during Superstorm Sandy.Before signing the Manhattan contract, Grant already had gained a measure of fame by, among other things, winning the Dr. …