unctuous puffing

Dear all
What's meant by "unctuous puffing" in the following context, taken from "The Three Soldiers" by Dos Passos.
Time: 1918
Location: a hospital somewhere in France

Andrews was roused suddenly from his thoughts; he had been watching the snowflakes in their glittering dance just outside the window pane, when the sound of someone rubbing his hands very close to him made him look up. A little man with chubby cheeks and steel-grey hair very neatly flattened against his skull, stood at the window rubbing his fat little white hands together and making with each breath.

I wouldn't have stopped at that point if I had been reading the book.
It's not that I have a clear understanding of "unctuous puffing" (I don't), but unctuous - oily, greasy, smooth, self-satisfied, complacent - seems to fit nicely with the little man, his chubby cheeks, his greased hair (otherwise it wouldn't be flat against his skull) and his fat little white hands.

Perhaps he makes a little noise in each breath, a small glottal stop followed by a slightly audible exhalation. I could make that seem unctuous 👁 Smile :)

Thanks dear Panj
It's not that I have fully grasped the picture but I'm trying my best to digest it, thanks to your explanations.