Make the dough: Lightly oil a large bowl with olive oil and set it aside. In the clean bowl of a stand mixer fitted with a dough hook, add the flour, 1 ½ cups tepid water, the salt, sugar, and yeast. 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. Place a pizza stone in the oven and preheat to 500°F.
Make the filling and topping: In a large bowl, combine the cheeses. In a small bowl, set aside 1 ½ cup of the mixture for topping (the rest will be used as the khachapuri filling).
Use a bench scraper or knife to divide the dough into five equal pieces (about 6 ounces each). Loosely shape each into a ball and cover with a clean towel to prevent from drying out.
On a lightly floured work surface using a rolling pin, roll out one ball of dough into a 10-inch round. Place a generous half cup of the filling mixture into the center of the round, then fold and stretch the edges up to meet at the center. Pleat and pinch the dough to create a tidy center knot. Press the bundle down gently and carefully roll it down to an even, 7-inch disk. Pop any pockets of trapped air with a skewer or the tip of a sharp knife and set the khachapuri back under the towel. Repeat with the remaining 4 disks and the rest of the filling, re-flouring the work surface as needed.
Heat a large griddle or cast iron skillet over medium-high heat until hot. Add as many khachapuri as will fit in a single layer without crowding or overlapping. Cook on one side until the breads are well browned and lightly charred in some places, 3-4 minutes. Using a wide spatula, turn and repeat on the remaining side.
Place the khachapuri on a pizza peal or on a cookie sheet with no sides. Evenly distribute the cheese topping over the khachapuri and gently slide each loaf onto the pizza stone without crowding. Bake 6-10 minutes, until the cheese is fully melted and browned in some places. Carefully remove from the oven and let cool slightly before serving.