In a large mixing bowl, combine the warm milk and yeast, whisking gently. Let the mixture sit for 5-10 minutes until it becomes fragrant and slightly frothy. Once the yeast has bloomed, whisk in the egg and sunflower oil until fully incorporated.

In a separate bowl, mix the all-purpose flour and salt. Gradually add the dry ingredients to the wet mixture, stirring with a wooden spoon or silicone spatula until the flour is fully absorbed, forming a rough dough.

Transfer the dough onto a clean work surface and knead for approximately five minutes until smooth and elastic. Place the kneaded dough into a lightly oiled bowl, cover with plastic wrap or a damp tea towel, and let it rise in a warm area for about 1 hour or until it doubles in size.

Preheat your oven to 220°C (425°F). In a small bowl, mix the grated Imeruli and Sulguni cheeses and set aside.

Punch down the risen dough gently and turn it out onto a lightly floured surface. Roll the dough into a circle about 35 cm (14 inches) in diameter and 5 mm (¼ inch) thick.

Spoon a line of cheese along the top and bottom edges of the circle, leaving a small border of about 2 cm (½ inch). Roll the edges tightly towards the center to create two parallel cheese-filled tubes. Pinch the ends together to form an oval shape with an open center, resembling an eye. Fill the center with the remaining cheese, ensuring it is evenly distributed but not pressed too firmly.

Place the assembled dough on a parchment-lined baking sheet. Lightly beat an egg and brush it over the exposed dough edges using a pastry brush. Bake in the preheated oven for 12-15 minutes, or until the crust is golden brown and the cheese is bubbling.

Remove the khachapuri briefly from the oven, add the egg yolk to the center of the melted cheese, and return it to the oven for 1-2 minutes until the yolk is slightly set. Once baked, immediately top the cheese with a dollop of butter.

Serve your Georgian cheese boat hot and enjoy the creamy, cheesy goodness.