Dermatologist Removed Fatty Tissue During Mole Removal

Suspicious moles should be removed completely. Punch biopsies usually will not give enough information, as the architecture of the complete lesion is crucial for the diagnosis many times.

The depression you were talking about very often results from the tension on the wound. Wound healing takes about 1.5 years. Only then the scar has matured. Your scar will also improve. Usually wounds in the face heal very well. 

There is fat under all skin. Whether or not the fat is removed depends on the type of biopsy performed and the reason(s) behind choosing that particular biopsy type. If melanoma was suspected, then a punch biopsy would have provided the most information as it can sample all layers of the skin and fat right under the skin. In your particular situation, it is difficult to advise what to do now without a physical examination or a photograph. Generally, if the depression is mild, there is a good chance it will become less shallow with time. If you are still unhappy with how it looks after a year, then a consultation with a good reconstructive surgeon (Mohs, plastic, etc) would be a good idea.

Fat is a protective layer underneath the skin.  So to answer your question, there is fat underneath all moles.  It is difficult to speculate about the fatty tissue "popping out" and causing a depresssion.  A picture may help.