Thanks for posting the second image. As it's a different hand, it doesn't help with this possible H.
I understand that you can't easily obtain another sample of the same hand.
Regarding our surname, I'm confident with the _ickore.
As to the initial, I believe I have seen H formed in this manner but unfortunately can't find an example.
It's not a G (of which there are two examples). It is somewhat similar to the E in Executrix, but I don't think it's an E.
Sorry I can't give a definite answer.
Also, I wouldn't necessarily conclude that the D stood for Doctor unless there were another reason to do so.