In the bowl of a stand mixer, add yeast, lukewarm milk and water, plus sugar. Stir and let the yeast proof for 15 minutes.
In a medium bowl, combine 2 cups flour, basil leaves, and salt. Whisk until blended and set aside.
When the yeast is foamy, add olive oil and dry ingredients. With the dough hook attachment, process on speed 2 for 2 minutes. Increase to speed 3 and mix for 5 minutes, adding a tbsp. of remaining flour at a time. When dough clings around the hook, increase to speed 4 and knead for 4 minutes – if it sticks at the bottom of the bowl, add another tbsp. flour.
Drop dough onto a lightly floured work surface, and with floured hands, pull under to form a ball. Place it in a greased bowl and toss to coat with oil.
Cover with a clean dish towel and transfer to a draft-free area. Let it rise for 1:30 to 2 hours or until it doubles in size.
Meanwhile, in a small bowl, add the 3 cheeses and stir well before seasoning with freshly ground mixed peppercorns; stir again and set aside until needed. After rising time is up, preheat oven to 450ºF.
Drop dough onto a lightly floured work surface. Form a ball and, using a bench scraper, divide it evenly into two pieces. Pull each piece under to form a ball.
Working with one at a time, roll out dough into a rectangle of 11-inches long by 7-inches wide.
Add 2/3 cup cheese mixture onto the dough, leaving an inch around the edges.
Roll up the widest sides, tightly over the cheese. Take the ends and twist them together to form a boat shape. Spread out the other 1/3 cup cheese mixture.
Carefully but quickly, lift the khachapuri onto a baking sheet lined with parchment paper; reshape the dough if needed and spread the cheese evenly. Repeat the same steps with the other dough ball.
Brush the edges with egg wash before transferring to the preheated oven. Bake for 13 to 15 minutes or until golden brown and cheeses are melted.
Remove from the heat and add an egg into the center of each khachapuri; sprinkle on some smoked paprika
Return the baking sheet back to the oven. Bake for 5 to 7 minutes or until the eggs are almost set.
Remove from the heat and allow to cool for a few minutes. Sprinkle on some finely chopped parsley and serve immediately.
