Removing a Black Spot After a Mole Removal

Sometimes if not all of the mole is removed, there is a chance for the pigmented cells to return.  I would recommend that you see a board certified plastic surgeon for an evaluation. 

Apparently when the dermatologist removed the mole there were still cells left behind. Sometimes if the cells aren't completely irradiated the mole will start to grow again. In this case you will need for the mole to be extracted more deeply to make sure every bit of the cells are gone. Just remember to have the extracted mole biopsied  to make sure it isn't cancerous. 

It sounds like you had a mole "removed" by shaving it flat against your skin, which is a perfectly fine way to start to check the mole by biopsy, and also to smooth the cosmetic appearance of the area. Sometimes this spot never turns dark, but sometimes, it may. This is not dangerous. In reality, a raised mole is spherical, half beneath and half above the skin surface. The nevus cells that live in the center of the mole are pigmented brown, so sometimes when the top half of a mole is removed, these cells are exposed and can leave a freckle. This is not dangerous, but if the flat freckle annoys you, you may have it excised (with stitches) by a qualified dermatologist, dermatologic surgeon, or plastic surgeon. The quality of the outcome will depend on the care and skill of the physician.

From your history I suspect that your dermatologist performed a shave excision. In this procedure the scalpel blade is cut parallel to the skin surface. Oftentimes, there are still nevus cells below the surface. With sun exposure (and certainly the nose would get this), these nevus cells are activitated to produce melanin which caused the black color to return.
An excision can be done in which the pigmented area is cut into deeply and the whole mole is taken out.
If this mole is less than 4 mm. the average dermatologist should be able to excise this quite nicely. Larger than this I would recommend a dermatologist whose practice emphasizes skin surgery or a plastic surgeon whose practice emphasizes skin surgery.
Also, make sure the specimen is sent off to pathology, preferably to the same path lab that read your original biopsy. They would have much easier access to the original material.
Good luck.