Place a pizza stone in the center of the oven and preheat to 500°F. Meanwhile, lightly oil a large bowl with olive oil and set aside.
Make the dough: In the clean bowl of a stand mixer fitted with a dough hook, add 2⁄3 cup tepid water, the flour, olive oil, 2 teaspoons kosher salt, the yeast, and the sugar..Mix on low speed until the dry ingredients are completely hydrated, 2-3 minutes. Increase the speed slightly and mix until a smooth, wet dough forms, 3-4 minutes more.
Transfer the dough to the oiled bowl and cover the bowl with a lid or plastic wrap. Set in a warm place until the dough is almost doubled in size, 50-60 minutes.
Make the filling: In a large bowl, combine the cheeses, herbs, coriander, and salt.
On a lightly floured work surface, turn out the dough. Divide into two roughly 6-ounce piece and round each piece gently. Cover the pieces loosely with plastic wrap and set aside for 15 minutes.
On a piece of lightly floured parchment paper, roll half of dough into a 10-inch circle about 1⁄8 inch thick. Spread a quarter of the cheese mixture (about 1 scant cup) over the dough leaving a 1⁄2-inch border. On one side of the circle, tightly roll the dough about a third of the way toward the center. Repeat on the opposite end, leaving a 2-3 inch wide space between the two rolls. Pinch the narrow ends of the rolls together and twist twice to seal, making a boat shape; place another quarter of the cheese mixture in the middle, packing down lightly.
Keeping the khachapuri atop the paper, gently slide onto a pizza peel or overturned baking sheet. Repeat with remaining dough and cheese. Set the khachapuri aside, uncovered, for 15 minutes, until slightly puffed.
Just before baking, brush the edges of the khachapuri lightly with olive oil, then slide the breads atop the paper onto the stone, spacing them at least 3 inches apart. Bake until the dough is golden brown and the cheese is melted, 16-18 minutes. Serve hot.