In the months before his pal Eric Adams became mayor and made him chief of staff, Brooklyn attorney Frank Carone was hustling to build up a company he’d created to provide cash advances to doctors awaiting potentially lucrative insurance payouts in no-fault car accident cases. Carone’s firm relied on a team of recruiters to find doctors who might sign up. Those on board included his longtime friend, Zhan “Johnny” Petrosyants, and two of Petrosy…