Learn Turkish: Hangimiz yeşil piyonu seçiyor? - Which of us is choosing the green pawn?

The word hangi means "which". By adding the first-person plural possessive suffix -miz (meaning "our"), it literally becomes "our which". This is the standard Turkish way to say "which of us" or "which one of us".

Could I say bizim hangimiz for "which of us"?

Yes, you can! Adding bizim (our) emphasizes the "us". However, because the -miz ending on hangimiz already tells us it means "of us", the extra pronoun bizim is usually left out.

If hangimiz means "which of us", why doesn't the verb use the "we" ending (seçiyoruz)?

Because the core subject is "which one", which is singular. Just like in English where we say "Which of us is choosing" rather than "are choosing", Turkish uses the third-person singular verb form seçiyor.

Why is there a -u at the end of piyonu?

The -u is the accusative case ending, which marks a specific direct object. Because someone is choosing a specific pawn (the green one), piyon needs this ending to show it is the thing receiving the action of the verb.