Step 1 Prepare the Dough: In the bowl of a stand mixer, combine the flour, yeast, sugar, and salt. Use a whisk to combine. Add the water and softened butter. Fit the mixer with the dough hook and mix on low speed for 2 minutes. Increase the speed to medium and knead for 5 minutes. Turn off the mixer and remove the bowl from the stand. Remove the dough hook and shape the dough into a ball in the bowl. Cover the bowl loosely with plastic wrap or a damp clean kitchen towel and let rise in a warm, draft-free spot for 2 hours, until doubled in size.
Step 2 Refrigerate the Dough: Line a baking sheet with parchment paper. Lightly flour a clean work surface and place the dough on top. Divide it into 4 equal pieces. Form each piece into a tight ball and place them on the baking sheet. Refrigerate for 1 hour or up to overnight. (If refrigerating overnight, wrap the dough balls in plastic wrap.)
Step 3 Prepare for Baking: Line two baking sheets with parchment paper. Preheat the oven to 400°F, arranging the racks in the top and bottom thirds of the oven.
Step 4 Shape and Fill: Lightly dust a large clean work surface with flour. Remove one ball of dough from the refrigerator and roll it into an oval about 10 inches long by 7 inches wide. Place on a prepared baking sheet. Spoon one-quarter of the cheese mixture onto the center of the dough, spreading it to within 1 inch of the edges. Pull the dough's edges up around the cheese, folding and twisting the ends to form a boat shape. Repeat with the remaining dough balls.
Step 5 Bake: Brush the exposed edges with egg wash and bake for 18 to 20 minutes, until lightly golden and the cheese is melted.
Step 6 Remove from the oven. Crack an egg into the center of each khachapuri (if the cheese has puffed up, leaving no room for the egg, push the cheese down with the tines of a fork to make room). Sprinkle each egg with a pinch of flaky sea salt.
Step 7 Increase the oven temperature to 450°F and bake for an additional 8 to 10 minutes, until the egg whites are opaque and the yolks are still runny.
Step 8 Remove the khachapuri from the oven and let them cool slightly. Serve warm, sprinkled with dill, if desired.
Step 9 Note: Feel free to bake 1 or 2 khachapuri at a time, leaving the remaining dough in the fridge until ready to use. In this case, place the oven rack in the center position to ensure even baking.
