Learn Arabic (Modern Standard): أأنت طبيب أم مهندس؟

Both أ and هل turn statements into yes/no questions. However, only أ can be used when giving someone a choice between two things. You cannot use هل when asking an either/or question like this.

I learned that "or" is أو (aw). Why does this sentence use أم (am)?

Arabic has two different words for "or". You use أم in questions where you are offering specific alternatives (Is it A or B?), and it almost always pairs with the question particle أ. The other word, أو, is used for regular statements.

How do you pronounce the beginning of أأنت?

You pronounce the two "a" sounds back-to-back with a slight catch in your throat between them, like "a-anta". The first أ is the question particle, and the second is the start of the pronoun أنت.

Could I use this exact sentence to ask a woman if she is a doctor or an engineer?

No, you would need to change the words to their feminine forms. You would say أأنتِ طبيبة أم مهندسة؟ instead. The pronoun changes to the feminine أنتِ (anti), and you add the feminine ending ة to both job titles.