Answer: Retained tick mouthparts: toothed hypostome and part of the basis capituli. Unfortunately, it is not possible to definitively identify the genus of the tick from the parts available.
The answer to the second question is more interesting: If you had received a phone call on how to best remove this object, what would have been your advice?
Most of you correctly responded that the best way to remove a tick is by the use of forceps. The CDC has an excellent web page with instructions for removing ticks that you can access HERE. Excision is not necessary (!) unless the tick is not readily recognized (I've seen an engorged tick mistaken as an infarcted skin tag). In certain situations, retained mouthparts might also be difficult to remove without excising the surrounding skin, although the excised tissue does not need to be large.
In this case, the diagnosis (retained tick mouthparts) was only suspected which may explain why such a large excision was performed.