The noun قَاضٍ (“judge”) has a stem-final yāʾ preceded by kasrah.
قَاضِي + يَ → قَاضِيَّ (“my judge”).
One yāʾ belongs to the noun; the other is the possessive pronoun, with assimilation. Contrast ظَبْيِي, where the stem-final yāʾ follows a consonant with sukūn.