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 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 very slightly and mix until a smooth, wet dough forms, 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 queso: In a medium saucepan set over medium-high heat, add the butter and chorizo. Cook until the butter has melted and the chorizo has begun to render and lightly brown, 4-5 minutes. Add the jalapeno and onion and cook till softened, 5-6 minutes. Stir in the cumin and cook until very fragrant, 1-2 minutes. Add the remaining 2 teaspoons flour and stir to coat the vegetables, then slowly add the milk while stirring to combine. Reduce the heat to medium-low and simmer until the sauce has thickened 1-2 minutes. Add the tomato, scallion, and cilantro, and bring the mixture to a simmer. Remove the pan from the heat and stir in 1 ½ cup of jack cheese and all of the cheddar, half a cup at a time, until melted and incorporated. Add the heavy cream and season to taste with kosher salt. Transfer the queso to a shallow dish to 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 queso 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 cheese mixture in the middle.
Gently slide the khachapuri on the paper onto a pizza peal or cookie sheet. Repeat with remaining dough and cheese, then top both with the remaining ¾ cup of jack cheese. 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.