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To make the dough: Heat the butter and milk together in a small saucepan or in the microwave until the butter melts.
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Place the sugar (or malt powder), coriander, and salt in a large bowl, and pour the hot milk over it, stirring to dissolve the sugar. Cool to lukewarm, about 100°F to 110°F.
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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Stir in the yeast and flour, mixing until a shaggy mass forms. Cover with plastic wrap or a reusable bowl cover and let rest for 10 minutes.
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After the rest, knead until smooth; 8 to 10 minutes by hand, or 6 to 8 minutes at medium speed in a mixer. Knead in an additional 1 to 2 tablespoons flour if the dough is uncomfortably sticky.
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Place the dough in a greased bowl, cover, and let rise for 1 to 1 1/2 hours, until it increases in size by at least one third.
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To make the filling: Place all the ingredients in a large mixing bowl and beat to combine, or pulse everything together briefly with a food processor; leave some bits of the cheese intact.
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To assemble: Line two baking sheets with parchment. Turn the risen dough out onto a lightly floured surface and divide it into four equal pieces, about 150g each.
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Roll each piece into an oval shape about 10" long and 6" wide. Cover with greased plastic wrap, and let rest for 15 minutes.
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Spoon one quarter of the cheese mixture (about 119g) into the center of each and spread to within 1" of the edges.
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Pull the dough's edges up around the cheese, folding and twisting the ends to form a boat shape.
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Cover and let rise for 20 minutes, or until puffy but not doubled. While the breads are rising, preheat the oven to 375°F with two racks toward the center.
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Brush the exposed edges of the khachapuri with the egg wash and bake for 15 to 20 minutes, until they feel set; they won't have started to brown nor will the filling be bubbly, but don't worry; they're going to continue to bake once you add their egg-and-butter garnish.
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Remove from the oven and use the back of a spoon to make an indentation about 3" across in the filling of each khachapuri. Crack an egg into each, place a teaspoon of butter on top, and bake for an additional 8 to 10 minutes (for soft-set eggs), or 12 to 15 minutes (for firmer eggs).
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Remove from the oven and serve warm. Garnish with additional fresh herbs, if desired.
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Storage information: Store any leftover khachapuri, well wrapped, in the refrigerator for up to five days.
