Get a passport renewed super expedited at a passport agency?

So i;ve got a flight on August 2 and I just realized my passport is expired. I went on travel.state.gov and they say that I can set up an appointment if my flight is within 14 days. The nextt available appointment is July 29(3 days before my flight). I was wondering if I do it this way (through the passport agency) how long does it take for them to have my passport ready? I;m am going overseas to visit my sick grandma-if that;s relevant at all.|||maybe -contact your embassy or passport agency to inquire.