Learn Turkish: Hâkim davayı ne zaman bitirmeliydi? - When should the judge have finished the case?

This is the core of past unfulfilled obligation! It combines the verb stem bitir- (to finish) with the necessity suffix -meli (should/must), the buffer letter -y-, and the past tense suffix -di. Together, -meliydi means "should have [verb]ed" or "was supposed to [verb]".

Why does dava (case/lawsuit) have the -yı ending in this sentence?

Because it's the specific direct object of the verb bitirmek. We are talking about "the case," not just any case, so we need the accusative case suffix. Since dava ends in a vowel, it takes the buffer letter -y- followed by the suffix -ı, making it davayı.

I've seen bitmek used for "to finish". Why do we use bitirmek here instead?

Bitmek is intransitive, meaning something ends on its own (e.g., "The movie finished"). Bitirmek is transitive, meaning someone finishes something (e.g., "The judge finished the case"). Since the judge is actively finishing a specific object (davayı), we must use bitirmek.

Why is there a circumflex (little hat) over the 'a' in hâkim?