Fitzwilliam D.
Yelp
It might well be selective attention, but it appears to me that the number of bakeries in Sheepshead Bay, Brighton Beach, and Midwood serving Georgian breads grows every few months.
Apani is a worthy recent addition to those - I haven't seen a Georgian bakery in the city make better use of their Facebook page (https://www.facebook.com/ApaniBrooklyn). They have an extensive selection of Khachapuri, and are perhaps the only Georgian bakery offering bacon as a filling, even naming their shop after the pastry dough version of the Khachapuri. They also have a yeast dough version with pork and onions: Kubdari.
I can't say that I'm a fan of their Imeruli that seems buttered to resemble a pizza (and expand the market to those unfamiliar with the charms of Georgian cheese breads), and my ambivalence is mainly a result of the heavier cheese filling inside (I doubt that it was the traditional, Sulguni). But, I have enjoyed the Megruli (w/double cheese) and the Penovani (pastry dough with cheese). The latter being my favorite item on their menu, as the many folds of thin, buttery pastry serve as an excellent vehicle to showcase the cheese.
There are a host of other items on the menu for fans of Georgian food, including the popular salads: Badrijani (eggplant with walnut), Pkhali (spinach with walnut), Lobio (red beans and onion stew), along with the soupy dumplings (Khinkali), Chicken Satsivi (with walnut sauce), Chakhokhbili (chicken stew with onion, tomatoes).
Service has been courteous and helpful - with the Ukrainian server even insisting that I carry the box of Khachapuri flat so as not to cause the cheese bread to lose its shape and become a crumpled mess. Had it not been for the heavier cheese, that warning would have been redundant, for I would otherwise polish off the bread between the shop and the subway station (maybe 40 steps away).