In a small bowl, whisk yeast into milk until dissolved. Set aside until slightly foamy and very fragrant, about 5-10 minutes. Whisk in egg and oil until well combined. Set aside.

In a large mixing bowl, whisk together flour and salt. Make a well in the centre and pour in wet ingredients. Using a wooden spoon or silicone spatula, stir until a dry, shaggy dough forms. Turn out onto a clean work surface and knead until the dough become smooth, supple and is just barely tacky, about 5-10 minutes.

Transfer to a lightly oiled bowl, cover with plastic or a damp tea towel and move to a warm place. Let rise until doubled in size, about one hour.

Gently punch down your dough and turn it out into a lightly floured work surface. Roll the dough in a circular shape until it reaches about 25 centimetres (10 inches) in diameter. Pile the filling into the centre, patting it into a disk and leaving a 5-centimetre (3-inch) perimeter of dough around the edges. Pleat the dough around the filling, pinching it to seal the filling completely.

Transfer to a parchment-lined baking sheet. Bake until cheese is melted and the khachapuri is puffed and lightly golden brown, about 10-15 minutes. Transfer to a wire rack and allow to cool for about five minutes before slicing and serving.