It's not very clear actually because there were no rules regarding that. (AFAIK) We only have records for official Kadis(Regional Judges) and Divan-i Humayun (higher court). There are cases seen at kadi or divan-i humayun where one side is Muslim and the other side non-Muslim.
They are both institutions of the state and since the Ottoman Empire was an Islam state, they were obviously Islamic. But, Divan-i Humayun's decisions were "Sultan's Word" so they had to be fair. The usual/accepted route is to go Kadi first and then you can appeal to Divan-i Humayun. But it's seen im records that sometimes non-Muslims directly went to Divan-i Humayun because they believed that they wouldn't be treated fair at Kadi.
They are both institutions of the state and since the Ottoman Empire was an Islam state, they were obviously Islamic. But, Divan-i Humayun's decisions were "Sultan's Word" so they had to be fair. The usual/accepted route is to go Kadi first and then you can appeal to Divan-i Humayun. But it's seen im records that sometimes non-Muslims directly went to Divan-i Humayun because they believed that they wouldn't be treated fair at Kadi.