Who are the doctors/institutions that perform surgery for thyroid cancer with hidden scars?

Thyroid: I'm not exactly sure what you mean by "hidden scars" but several types of physicians might be involved in the care of thyroid cancer, including an endocrinologist, head and neck surgeon (or general surgeon), and an oncologist. There are several good options for thyroid cancer treatment and you should seek one that you feel the most comfortable with.

Da Vinci Robot: The procedure you are referring to is performed via small incisions in the armpit utilizing the da vinci robot (intuitive surgical). There are numerous hospitals across the country that offer this. If you try an internet search in your area, I am sure you can find a nearby surgeon.

Almost any surgeon..: Who does thyroid surgery will locate the incision in a crease at the base of the neck. Because this lies in a natural crease, it will tend to heal with a very fine scar.

Da Vinci Robot: You are likely referring to trans-axillary robot assisted thyroidectomy, done with da vinci robot. It is a new technique that is safe, but expensive. It is not for people with known thyroid cancer, large goiters, or obese patients. Seek out a teaching institution (usually ENT dept) with experience. For the record, most thyroid surgery can be done with a small incision which heals beautifully.

Thyroid scars: All well-trained thyroid surgeons will "hide" the scar in a natural skin crease of the lower neck. Even if you don't have a crease, it will be hard to see the scar so placed. Your best bet is to find a well-trained thyroid surgeon. Ask your family doctor for a referral.

Scarring: The chinese have developed an approach through the axilla (armpit). There are some institutions in the U.S. That do this. However, the scar in small and well concealed in the majority of patients. However, i think you will have to "google" this to get adequate satisfaction.