In a small bowl, dissolve the yeast and sugar in warm milk. Set aside for 5-10 minutes until frothy.
Into a large bowl, sift flour. Add salt, egg yolk, and the proofed yeast. Using a wooden spoon, mix everything until combined. Once the dough ball forms, turn it onto a clean work surface.
Knead the dough for 3-5 minutes until a smooth, elastic dough ball is formed. Into a large bowl, pour in a little oil and oil the bottom and sides of the bowl. Place the dough ball into the greased bowl and roll it around to cover the dough ball in oil. Cover with a dry tea towel and set aside for 1 hour in a warm, draft-free place.
After the dough rises for an hour, knead the dough in the bowl for 20-30 seconds. Cover the dough with a tea towel and let it rise for another 30 minutes.
As the dough rises, let's prepare the filling. Grate the sulguni cheese or cheese of your choice, combine it with 1 tbsp of milk, and mix it. Set aside.
Line two baking sheets with parchment paper and preheat the oven to 450 degrees F. After the second rise, turn the dough onto a floured surface and divide it into four equal parts. Roll the dough into an oblong shape about 10 inches by 6 inches using a rolling pin. Aim for a 2-3 cm thickness.
Place the oblong dough shapes onto the prepared baking sheet. Spread the cheese over it, leaving a 1-inch border. Pull the edges of the dough toward the center and roll just until it meets the cheese filling. Take the corners of the dough, twist, and pinch to form a boat shape.
Place the khachapuri into the oven and bake until golden—about 15 minutes.
Remove the khachapuri from the oven, and in the cheese, make an indentation with the back of a spoon. Crack an egg into the indentation and place it in the oven for 2-3 minutes. Before serving, place a small slice of butter in each khachapuri.
