here are several types of Khachapuri one might encounter throughout Georgia. There is Imeruli, Megruli, Guruli, Osuri, Svanuri, Rachuli, Penovani, etc. Each one of these are made in slightly different variations. Adjarian khachapuri will also remind you of Turkish Pide. Turkish pide can be made with lots of savory toppings, where Khachapuri is strictly a cheese and a thicker crust thing. Some variations can be made with the addition of an egg.
If you would like to try cracking an egg on the khachapuri, take the baking sheet out of the oven right before the khachapuri is done baking. Crack one egg right in the middle of it and place the khachapuri back in the oven until completely baked.