For dough: In the bowl of a stand mixer, whisk together 4 cups (508 grams) flour, sugar, yeast, and salt by hand. Using the paddle attachment, with mixer on low speed, add 1 cup (240 grams) warm water, warm milk, and oil, beating until a shaggy dough forms.

Switch to the dough hook attachment. Beat at medium-low speed until dough is smooth, elastic, and slightly tacky, 12 to 15 minutes; add up to remaining 1⁄3 cup (42 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky.

Lightly oil a large bowl. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, 45 minutes to 1 hour.

For filling: In a medium bowl, stir together mozzarella, feta, and crème fraîche. Stir in egg yolks, chives, thyme, salt, and pepper. Refrigerate until ready to use.

Line 2 to 3 baking sheets with parchment paper; lightly dust parchment with flour.

Punch down dough, and let stand for 5 minutes. Turn out dough onto a lightly floured surface. Divide dough into 4 portions (about 222 grams each) for larger khachapuri or 6 portions (about 148 grams each) for smaller khachapuri.

Shape 1 portion into an oval and then press flat. (Keep remaining portions covered to prevent them from drying out.) Roll dough into a 10x6-inch oval for larger khachapuri or 9x6-inch oval for smaller, flouring surface as needed. Repeat with remaining dough. Cover and let stand for 15 minutes.

Working with 1 portion of dough at at a time, roll edges of dough toward center on each side to form a border to hold filling. Using fingertips, press rolled edges of crust together with base of dough to seal well. (This will prevent moisture from the filling from unraveling the edges while baking.) Pinch ends together, then twist and slightly tuck under to form boat shape. Place on a prepared pan. Repeat with remaining dough, placing 2 larger or 3 smaller on each prepared pan.

Divide filling among dough, 1¼ cups (about 257 grams) each for larger dough or 2/3 cup (about 174 grams) each for smaller, and spread evenly within border. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed (not doubled), about 30 minutes.

Preheat oven to 375°F (190°C).

In a small bowl, whisk together egg and 1 tablespoon (15 grams) water; using a pastry brush, brush onto dough. 12. Bake until crust is golden brown, 17 to 19 minutes for smaller or 20 to 23 minutes for larger. Add 1 egg yolk to center of each khachapuri, and bake until edges of yolks just start to set, about 2 minutes more. Top each with 1⁄2 tablespoon (7 grams) butter. Garnish with thyme, chives, and pepper, if desired. Serve immediately.