Learn Arabic (Modern Standard): ما مهنتك، أأنت مهندس أم طبيب؟ - What is your profession, are you an engineer or a doctor?

In Arabic, أم is specifically used in questions where you are asking someone to choose between specific options (an alternative question). Since the speaker is asking the listener to choose "engineer OR doctor?", we must use أم. The word أو is generally used for "or" in regular statements.

What is the extra letter at the beginning of أأنت?

The first أ is the question particle أَ (a), which works like هل (hal) to turn a statement into a question. The rest of the word is the pronoun أنت (you). When asking an alternative question with أم, it is standard grammar to start the question with أَ rather than هل.

Why do we use ما instead of ماذا for "what" in the first part of the sentence?

We use ما because it is followed by a noun, مهنتك (your profession). The other word for "what", ماذا, is only used when it is followed by a verb.

How can I tell if this question is being asked to a man or a woman?

In written Arabic without short vowel marks, it looks exactly the same for both! If you are speaking to a man, you pronounce the endings with an "a" sound: mihnatuka and anta. If speaking to a woman, you use an "i" sound: mihnatuki and anti.