“Private kitchen” = 私房菜 = home-cooked or super-traditional meals that are fixed course depending on the day and the restaurant is usually set in a small apartment upstairs; was all the rage in Hong Kong some years ago.
If you’re in Hong Kong, why don’t you try a few set up by Mr. Lau Kin-Wai:
Yellow Door
Kin’s KitchenI went to the Yellow Door some years ago and both the setting and food were unique experiences 
Via Thomas Crampton.
There are many private kitchen around mainland as well, but the business seems not tat good as couple yrs ago