What is the best way to remove flat, dark moles from the face?

In my opinion and long experience, Scalpel Sculpting followed by Dermaplaning (if needed) would be the treatment of choice for removing flat moles--dark or lighter-colored--from any location, including the face.Although removing moles by any method from the face is likely to leave a small scar, scalpel sculpting, which involves no deep cutting or stitches has, in my experience, proven quite successful for achieving gratifying aesthetic results while leaving little, or often barely perceptible, scars.The technique, which I have been using for thirty years, involves "scultping the mole" off from the surrounding skin in a tangential fashion (i.e. not cutting deeply into the skin). Deep cutting will inevitably result in a scar, while superficial (horizontal) removal in this fashion largely avoids this. Elliptical and fusiform simply describe the resulting shape of a wound excision after cutting them out deeply and before the placement of the sutures.As an important aside, destructive modalities to simply destroy the mole, e.g. lasers, electrocautery, electrodessication or cryosurgery should not be performed since these simply destroy all the mole tissue and do not permit a small specimen to be sent to the lab to ensure that the mole removed was entirely benign.If necessary, following scalpel sculpting, the borders of the mole can then be smoothed and blended with the surrounding normal skin by "dermaplaning," a technique by which the edge of the scalpel is used to delicately abrade the skin. Properly done, the entire procedure, performed under local anesthesia, takes no more than three to five minutes. In most cases, the procedure is done at the time of the consultation.               

How you treat pigmented lesions on the face depends on what type they are. Your description fits better with flat "seborrheic keratoses" than with "moles" - also called "nevi". Keratoses  can be treated by a laser or by freezing or  with a chemical. Nevi ought to be excised rather than destroyed to get a path report. An experienced physician can often tell what type your lesions are without a biopsy - but not always. 

Facial moles can be easily removed using a shave excision
technique that typically results in a minimal scar. With this method, the
mole is actually shaved off at a very superficial level just below the surface
of the skin. The procedure typically takes less than a minute and is painless
after a tiny injection to numb the area. Moles that are removed in this
fashion should not recur.