In the bowl of a stand mixer, whisk together 4 cups (508 grams) of bread flour, 1 tablespoon (12 grams) of granulated sugar, 2¼ teaspoons (7 grams) of instant yeast, and 2 teaspoons (6 grams) of kosher salt by hand. This mixture forms the foundation of your dough.
Using the paddle attachment, with the mixer on low speed, add 1 cup (240 grams) of warm water, ½ cup (120 grams) of warm whole milk, and 2 tablespoons (24 grams) of olive oil. Beat until a shaggy dough forms, which means the ingredients are just starting to come together.
Switch to the dough hook attachment. Beat at medium-low speed until the dough is smooth, elastic, and slightly tacky, about 12 to 15 minutes. If the dough seems too sticky, add up to an additional 1/3 cup (42 grams) of flour, 1 tablespoon (8 grams) at a time.
Lightly oil a large bowl and place the dough in it, turning to grease the top. Cover the bowl and let it rise in a warm, draft-free place (around 75°F/24°C) until it doubles in size, which will take about 45 minutes to 1 hour.
For the filling, stir together 6 ounces (340 grams) of shredded mozzarella cheese, 6 ounces (340 grams) of crumbled feta cheese, and 1 cup (320 grams) of crème fraîche in a medium bowl. Fold in 2 large egg yolks (37 grams), 2 tablespoons (8 grams) of chopped fresh chives, 4 teaspoons (4 grams) of chopped fresh thyme, ½ teaspoon (1.5 grams) of kosher salt, and ½ teaspoon (1 gram) of ground black pepper. Mix thoroughly and refrigerate until ready to use.
Prepare baking sheets by lining 2 to 3 of them with parchment paper, lightly dusting it with flour to prevent sticking.
Once the dough has risen, punch it down gently and let it stand for 5 minutes. Turn the dough onto a lightly floured surface and divide it into 4 portions (about 222 grams each) for larger Khachapuri or 6 portions (about 148 grams each) for smaller versions.
Take one portion of dough and shape it into an oval, pressing it flat. Keep the remaining portions covered to prevent them from drying out. Roll out the dough into a 10x6-inch oval for larger Khachapuri or a 9x6-inch oval for smaller. Repeat this process with the remaining dough, covering it and allowing it to stand for 15 minutes.
Focusing on one portion of dough at a time, roll the edges toward the center on each side to create a border to hold the filling. Gently press the rolled edges of the crust together with the base of the dough to seal it well, preventing moisture from escaping during baking. Pinch the ends together, twist, and tuck them slightly under to form a boat shape. Place this on a prepared pan. Repeat with the remaining dough, ensuring that you place 2 larger or 3 smaller on each prepared pan.
Divide the filling evenly among the dough pieces, using 1¼ cups (about 257 grams) each for larger dough or 2/3 cup (about 174 grams) for smaller. Spread the filling evenly within the border of each piece. Cover and let rise in a warm place until puffed (not doubled), about 30 minutes.
Preheat the oven to 375°F (190°C).
In a small bowl, whisk together 1 large egg (50 grams) and 1 tablespoon (15 grams) of water. Using a pastry brush, coat the dough with this mixture.
Bake the Khachapuri until the crust is golden brown—17 to 19 minutes for smaller ones and 20 to 23 minutes for larger. Add 1 egg yolk to the center of each Khachapuri and return to the oven until the edges of the yolks just start to set, about 2 minutes more. Top each with ½ tablespoon (7 grams) of butter. Garnish with additional thyme, chives, and pepper if desired. Serve immediately to your eager guests!
