The Adam’s apple is an important gender sign.
Despite what many people believe, it is sometimes visible in women, but in men it can be very prominent. The main problem area is a "V" shaped structure called the "laryngeal prominence". You can see it in this illustration:
The Adam’s apple can be reduced with a procedure often referred to as a “tracheal shave” or “trach shave”. The technical name for this procedure is “chondrolaryngoplasty” but you may also hear of it referred to as “thyroid cartilage reduction” or “TCR” for short.
The incision is sometimes made in the neck directly over the adam's apple. The problem with this approach is that it leaves the scar in a very visible place and it can also cause adhesions where the scar tissue sticks to the Adam's apple and moves with it.
Some surgeons will tell you that the scar will be invisible. This is not true. The photo below (that someone has been kind enough to let me use) shows a typical example of one of these scars. Some are better than this but many are much worse. This patient also had adhesions.
The more advanced approach is to make the incision under the chin near to the area where the chin meets the neck and in a natural crease of the skin if possible. The surgeon then tunnels down to the Adam's apple under the skin. The advantage of this is that the scar is much less visible because it's hidden under the chin and you don't have the risk of adhesions.
Bear in mind that the surgeon does not have complete control over the scar - some people scar more easily than others and some people might scar lightly on one occasion but badly on another. My advice is not to take the risk - find a surgeon who puts the incision under the chin.
The cartilage that forms the Adam’s apple tends to harden with age and that may limit the amount that can be removed in an older patient. There is also a limit to how much can be removed without risking permanent damage to the voice. Basically, the Adam's apple is part of the front wall of the voice box and the vocal chords are attached to that wall - if the wall is thinned too much, it may no longer be strong enough to hold the vocal chords in full tension. This can leave the voice deeper and with a gravelly sound.
Dr Spiegel has written an article on the procedure that you can read here.
And you can see Facial Team's 3D animation of the procedure here.