The problem with #1 of course is "how did I decide what type of specialist I need"? I started by looking for "sleep", not expecting that you had subdivided into different specialists within sleep. Great. Ok, I now have a list of specialists, but how did I know that I needed a specialist in Central Apnea, as an example. If I am qualified to no the specifics of a disease, I'm probably already speaking to a doctor, so ....
Are other doctors the target market? If so, is this the right UI for them? Or would a "LinkedIn for medical" be a better experience, allowing doctors to "connect", review/recommend each other, etc etc
The current UX doesn't actually give any usable info on the doctor, aside from their contact details, on the searches I've looked at. I have no idea what your "Expert Physician", or "Expertise Team" rankings mean. Or the "Clinical Trials" phases. There are no links for me to do anything.