Alert! Have recently gone to Famous Pizza & Souvlaki. It's at 10 N. Water in the Byram section of Gwich.
It's small and looks like any neighborhood pizza joint, but the Greek side of the menu has a handful of items and is superb. I learned the chef/owner is a CIA grad.
Sharing a few dishes, we had a Greek dip assortment. Delicious warm pita (not greasy); eggplant, tzasziki, taramousolata and a fabulous feta & hot pepper. We shared a very fresh salad with dill and feta and a combination platter entree: heaps of gyro, souvlaki, falafels, pita, etc.
Good baklava for dessert (several other selections looked tasty), served with fresh honey -- not the thinned down honey syrup stuff.
3 people happily stuffed for $35. You can get good falafel or gyro sandwiches for $4.50.