Is it permissible for a woman to specialize in general surgery to treat women, in order to avoid exposing their awrah (private parts) to male doctors, even though this specialization requires mixing with male doctors?

If your intention in specializing is to examine women instead of having them exposed to examination by men, while maintaining modesty and avoiding mixing as much as possible, then there is no sin upon you in pursuing this specialization. Rather, you will be rewarded for it. This is because the harm of mixing, which you may be compelled to do, is far less than the harm of men seeing women's private parts. The Sharia (definition) is based on achieving benefits and reducing harms, and on preferring the better of two good things and warding off the worse of two bad things.