I looked up Korney Chukovsky then did an image search and saw pictures from his books of a man holding a rabbit with a cross on his hat and found his name "Aybolit". Then I looked up Doctor Aybolit and guessed it was him. 
It's getting more mysterious.
Why did you watch Korney Chukovsky?
It seems that I called this name after you guessed the character. Or have I let it slip somewhere before?
(First - I am having a few rounds of immunotherapy for cancer, (prevention) which leaves me a bit unwell but well enough to sit in front of a computer for a few hours and keep looking for things until I find them. I like solving puzzles, and I had to solve one where you told me I wouldn't know who the character was... That meant I HAD to find who it was. )
I did a search in Google something like; 'russian literary character with hat animals doctor'. Eventually I found this;
https://en.wikipedia.org/wiki/Doctor_AybolitThe picture at the top looked a bit like the man in the drawing, but I still wasn't 100% positive, so I used the authors' name 'Korney Chukovsky books' to do an image search and found a picture of Doctor Aybolit with the little white hat with the cross on it, then did an image search of Doctor Aybolit.
The internet knows everything.