I used the basic flatbread recipe a while back and added all my holiday leftover heavy cream and melted butter. It was excellent! It will crack when cold and with these two ingredients, it will stick together when it gets cold and is stacked. (you have to heat the hole pile to get one!)
For making gyro's I would definitely use oil which will not set when it cools.
And I would probably avoid milk too....definitely the heavy cream.
You may also want to use a lower heat and cook them until only lightly browned. The inside should not be too dry.
Flatbread and brats are a favorite here.
"I have not failed. I've just found 10,000 ways that won't work." --- Thomas Edison