Place a pizza stone in the center of the oven and preheat to 500°F. Lightly oil a large bowl with a drizzle of olive oil and set it 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, 1 1⁄3 cup flour, and the salt. Mix on the lowest speed until the dry ingredients are completely hydrated, 2-3 minutes, then increase the speed slightly and mix until a smooth, wet dough comes together, 3-4 minutes. Transfer the dough to the oiled bowl and cover with a lid or plastic wrap. Set in a warm place until just about doubled in size, 50-60 minutes.
Make the buffalo chicken filling: Season the chicken breast with salt. In a medium saucepan heat 1 tablespoon of butter over medium-high heat. When the foam begins to subside, add the chicken breast to the pan. Cook, turning as needed, until the chicken is just cooked through, 5-6 minutes per side. Remove the chicken from the pan and set aside. Turn off the heat but do not wash out the pan. Let the chicken cool slightly, then shred.
In the same pan over medium-low heat, add the remaining 4 tablespoons butter and let melt, using a whisk to scraping up the browned chicken bits from the bottom of the pan. Whisk in the hot sauce, vinegar, and Worcestershire sauce, then return the cooked chicken to the sauce. Remove the pan from the heat and stir in 1 cup of pepper jack until melted (mixture will not incorporate into a smooth sauce). Transfer the buffalo chicken mixture to a shallow dish and let cool to room temperature.
Shape the khachapuri. Lightly flour a clean work surface and turn the dough out onto the counter. Divide the dough into two roughly 6-ounce pieces and round each piece gently. Cover the pieces loosely with plastic wrap and set aside for 15 minutes.
On a sheet of lightly floured parchment paper, roll half of dough into a 10 inch circle about 1⁄8 inch thick. Spread a quarter of the chicken mixture over the dough, leaving a 1⁄2 inch border all the way around. On one side of the circle, tightly roll dough about a third of the way toward the center. Repeat on the opposite end. There should be a 2-3 inch 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 chicken mixture in the middle.
Gently slide the khachapuri onto the paper onto a pizza peal or cookie sheet. Repeat with remaining dough and cheese, then top both with the remaining 2⁄3 cup of pepper jack. Set the loaves aside, uncovered, for 15 minutes, until slightly puffed.
Immediately before baking, brush the edges of the khachapuri lightly with olive oil, then slide them on paper onto the stone, at least 3 inches apart. Bake until golden brown, 15-17 minutes.