Place a pizza stone in the center of the oven and preheat to 500°F. Meanwhile, lightly oil a large bowl and set aside.
In the bowl of a stand mixer fitted with a dough hook, combine the yeast, sugar, olive oil, and 2⁄3 cup tepid water, the flour, and salt. Mix on the lowest speed until the dry ingredients are completely hydrated, 2–3 minutes, then increase to medium-low speed and mix until a smooth, wet dough comes together, 3–4 minutes.
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 has almost doubled in size, 50–60 minutes.
Make the filling: In a large bowl, combine the cheeses and set aside.
Crack the eggs gently into a separate small bowl (if the yolks break, start over).
Onto a lightly floured work surface, turn out the dough. Divide into two roughly 6-ounce pieces and round each piece gently. Cover 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 5 ounces) over the dough, leaving a 1⁄2-inch border all the way around. 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 two 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 (with the paper) onto the stone, spacing them at least 3 inches apart. Bake until the dough is light golden brown and the cheese is melted, 14–16 minutes. Open the oven door and gently pour 1 egg into the center of each boat, being careful not to bread the yolks. Close the oven and continue baking until the whites are just set, 3–4 minutes.
Remove from the oven and dollop the butter evenly on top of the two breads. Serve hot.