Khachapuri (Georgian Cheese Bread)

To make the dough: Heat the butter and milk together in a small saucepan or in the microwave until the butter melts., 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., Stir in the yeast and flour, mixing until a shaggy mass forms. Cover and let rest for 10 minutes., 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., 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., 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., 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., Roll each piece into an oval shape about 10" long and 6" wide. Cover with greased plastic wrap, and let rest for 15 minutes., Spoon one quarter of the cheese mixture (about 119g) into the center of each and spread to within 1" of the edges. Pull the dough's edges up around the cheese, folding and twisting the ends to form a boat shape., Cover and let rise for 20 minutes, or until puffy but not doubled. While the breads are rising, preheat the oven to 375°F., Brush the exposed edges of the khachapuri with the egg wash, and bake for 15 to 20 minutes., 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 10 to 12 minutes more, until the egg is set., Remove from the oven and serve warm., Store any leftover khachapuri, well wrapped, in the refrigerator for up to 5 days.