In the bowl of a stand mixer, combine lukewarm water and milk. Add in yeast, sugar, and salt. Stir to combine. Let it rest for a few minutes or until bubbly.
In small parts, add flour to the liquids. Using a dough hook, knead on low speed as you add in flour. Add oil to the mixture. Knead until the dough is elastic and no longer sticky.
Cover the mixing bowl and let the dough rise for about an hour.
KHACHAPURI FILLING
In a large bowl, combine the cheeses together. Stir until the mixture is perfectly smooth.
Divide the dough into 6 equal parts. Flour the working surface, then roll the dough out into a round shape.
Fill the center with 1/2 cup of cheese. Pull the edges of the dough together, pinching tightly.
Dust generously with flour and lightly flatten the khachapuri with your hands, followed by the rolling pin. They should be about 6 inches in size.
Preheat a pan on medium heat with a tiny bit of oil. Cook khachapuri on each side for about 4 minutes or until golden.
Remove from heat, brush with butter. Serve right away!