The white stuff is plain yogurt I believe, we used to do gyros once a week when I lived in Houston. MAN!! I miss those.
It's Tzatziki sauce, yogurt and a bunch of spices. So good! Will make you slap your mama.
My wife dragged me to a local Greek place in Lubbock where I tried to order a burger. The guy talked me into a giro, said if I didn't like it he'd refund my money and make me a burger. I ended up ordering another after I knocked off the first one. They use a combination of lamb and beef.
Tzatziki Recipe