Very interesting. I can see that "Γιατρέ μου" wouldn't work.
I know this may be subjective but am I right @Zazula (or anybody else) in saying that "η γιατρίνα, η γιατρέσα, η αρχιτεκτόνισσα & (προφ.) η μηχανικού" are awkward and rarely used?
As I said in your question about the architect, as a rule women professionals of traditionally male professions like archtect, lawyer, or doctor, do not like to refer to themselves by the female type. One reason is that all of these were meant in the past for the wife of the professional, e.g. γιατρίνα. So anything that ends in -ίνα rings silly. Female lawyers for instance find δικηγορίνα degrading. In an English-language equivalent, it is like calling them "little lawyer".
In other cases, the suffix has a bad connotation, e.g. γιάτρισσα (see μάγισσα), i.e. anything that ends in -ίσσα. Γιάτρισσα is fine in literature, or metaphorically, but it mostly alludes to the notion of a self-trained practical "healer".
Anything ending in -ού is used colloquially (e.g. η νυχού, η λουλουδού) and I would not recommend using it for your doctor or engineer.
There are some efforts to promote widespread use of female types, but unfortunately they don't seem to be catching on yet.
You cannot go wrong wıth γιατρός, γιατρέ (μου).