In a small bowl, combine the milk, sugar, and yeast. Allow to rest for 10 minutes, until frothy.
In the bowl of a stand mixer fitted with the dough attachment or a large bowl, combine the yeast mixture with the yogurt, egg, sour cream, salt, and bread flour to create a smooth and slightly sticky dough.
On a lightly floured surface, knead the dough to form a smooth and elastic ball, about 4-5 minutes. Transfer to a lightly oiled bowl, turning to coat, and cover. Allow to rest in a warm spot for 75-90 minutes, until puffed.
To make the filling:
Place the unpeeled potatoes in a pot and cover with water. Bring to a boil, add salt, and reduce heat to medium low and cover. Cook until the potatoes are tender. Drain and allow to cool enough to handle.
Peel the potatoes and grate or mash into a mixing bowl. Fold in the melted butter, feta, and mozzarella (don't overmix). Season to taste with salt and pepper.
Preheat oven to 400˚F (200˚C). Place a pizza stone or large round baking sheet on the center rack of the oven.
On a lightly floured surface, roll the risen dough into a large, thin circle. Place the filling in the center, spreading it out into a flat layer, leaving a large border around the edges.
Fold the edges in towards the center and seal the edges well to completely cover the filling. Flip the bread over, seam-side down, and gently pat or roll to create a smooth, stretched circle with the filling sealed inside.
Pierce the top of the bread all over with a fork and brush with the beaten egg.
Transfer to the stone or hot pan in the preheated oven and bake until golden, 30-40 minutes. The bread may puff, but will settle back down when removed from the oven.
Immediately brush the top with butter. Cut into wedges and serve hot.