I have never tried to make Gyro's. But, I love them. Unfortunately I am in the same situation as HomeBrew. No Gyro's around locally. It is also a 100 mile drive for me to get one, so only get them when I am in the 'big city'. Either Portland or Eugene.
I have raised lamb though, for many years. Usually get 2 or three in early spring. Keep them until fall and then butcher. Since mine are always youngsters, there is never any strength to them. Never any 'mutton' taste or smell, as there sometimes is on older animals. Just to defend against that possibility, however, I always trim every bit of fat I can find.
If I want the meat for the main ingredient in some dish, I cook 'low & slow' to make sure all remainimg fat gets rendered out. The fat is usually the culprit for the 'mutton' taste. I treat it almost like 'pulled pork', when trying to achieve this situation.
If I am working with chops or steaks, I will run smoke against them, but then cook them on the grill or under the broiler. Here I treat them much like a pork shoulder steak. I cook at a higher temp and faster, searing the meat and locking in the juices.
I even grind the trimmings and use them straight to make lamb burger. Or sometimes mixed with a good ground beef, to make up a hybrid lamb/beef burger. This hybrid is the combination I used in my recent post about 'smoked meatloaf'.
I personally think that lamb takes very kindly to smoke and is really worth trying, for those of you who have not given it a shot.