For the Dough:
Heat the butter and milk together in a small bowl in the microwave until the butter is melted. Place the sugar, Indian coriander, and salt in a large bowl or mixer with hook attachment. Add the hot milk and stir to dissolve the sugar. Let the temperature comes down to warm.
Stir in the yeast and 2-3/4 cups of the flour until a loose ball forms. Cover and let the dough rest for 15 minutes.
After resting knead the dough until smooth. By hand, it will take 10 minutes or 7 minutes with a mixer on medium speed.
Place the dough in a greased bowl, cover and let rise for 1-1/2 hours, until it grows in size by a third.
For the Filling:
Place all the ingredients in a large mixing bowl and mix to combine. Line two baking sheets with parchment.
For the Assembly:
Preheat the oven to 375 degrees.
Add the dough out onto a lightly floured surface and split it into four equal pieces.
Roll each piece into an oval shape about 10 inches long and 6 inches wide.
Spoon one quarter of the cheese mixture into the center of each and spread to within 1 inch of the edges. Pull the dough's edges up around the cheese, folding and pinching the ends to form a boat shape. If you're good at crimping, now is the time to show off your skills.
Cover and let rise for 20 minutes, or until puffed. Brush the edges with the egg wash and bake for 20 minutes.
Remove boats from the oven and use a spoon to make an indentation in the cheese, large enough for the egg. Crack an egg into each hole, place a teaspoon of olive oil on top, and bake for 10 to 12 minutes more, until the egg is set to your liking.
Remove from the oven and serve hot.
Notes:
Feel free to sprinkle Marash Chile Flakes on the Khachapuri for some extra heat.