Georgian Cheese Boat Breads (Adjaruli Khachapuri)

INSTRUCTIONS:
You will need a large bread bowl, a medium-sized bowl, unglazed quarry tiles to fit on a rack in your oven, a baker's peel or two baking sheets and a rolling pin (optional).
Place the warm water in a large bowl, stir in the sugar and the yeast, and let stand for several minutes until the yeast has dissolved.  Then gradually add 2 1/2 cups flour, stirring constantly in the same direction, about 1 minute, to develop the gluten.  Sprinkle on the salt, add the oil, and continue adding the flour and blending it onto the dough until it is less sticky.
I've taken to using one of my silicone spatulas instead of a wooden spoon for these jobs - it seems to stick less and mix more.  It could just be a psychological thing too:

Turn the dough out onto a floured surface and knead for 10 minutes, until smooth and elastic, with a slight sheen. Form onto a ball, and place in a lightly oiled clean bowl or on a lightly floured surface to rise, covered with plastic wrap, until doubled in volume, 1 1/2 to 2 hours.
This was my rise after an hour:

Preheat the oven to 450 degrees.  If using quarry tiles, arrange on the bottom oven rack, leaving a 1-inch gap between the tiles and oven walls.  If not, lightly oil two baking sheets.
Gently push down the dough.  On a lightly floured surface, using a sharp knife, cut the dough into 4 equal pieces.  Flatten each piece out with the lightly floured palm of your hand, then cover with plastic wrap while you prepare the filling.
Since I'd halved the recipe, I only had two pieces - that's a standard cookie sheet that they're resting on at the moment.  I should also note that I have a round pizza stone - I had the stone in the oven getting hot while the oven preheated:

Place the cheeses and yogurt in a bowl, and blend together to a smooth consistency.
This is very straightforward (although "smooth" is relative since you're supposed to start out with coarsely grated Gruyere).  What I will say is that for the bite that each of these ingredients has separately, they come together and seem almost sweet.  Husband said it almost reminded him of cheese danish filling.

Work with one piece of dough at a time, leaving the remaining dough covered with plastic wrap.  With your hands or a rolling pin, stretch and flatten the dough in to a long oval 8 to 10 inches long, 5 to 6 inches wide, and no more than 1/4 inch thick.  Place a generous 1/4 cup filling in the center of the oval.  Spread to within an inch of the edges.  Roll the edges over to make a thick rim, pinching the sides together to form a point at the ends.  (The bread should look boat-shaped.)  Shape and fill a second bread.  Slide the breads onto a peel and then onto the quarry tiles, or slide onto the baking sheets and place on the bottom oven rack.  Bake until the crust is golden and the bottom is firm and crusty, about 12 to 15 minutes.  Wrap in a towel to keep warm while you prepare and bake the remaining two breads the same way.  Serve hot.
The construction took less than 5 minutes per boat:

I only did one boat at a time, as my stone is smaller.  The bake time was about 14 minutes for each.  The puff was gorgeous: