In a large mixing bowl, combine warm water, warm milk, granulated sugar, and active dry yeast; stir gently and let it foam for 8 minutes. In a separate bowl, mix the all-purpose flour and salt, then pour in the yeast mixture, yogurt, a beaten egg, and melted butter. Mix at low speed, then increase to medium until the dough is smooth and elastic, about 10 minutes.

Transfer the dough into a lightly oiled bowl and cover it with plastic wrap. Let it rise in a warm, draft-free area for about 1 hour and 15 minutes, or until it has doubled in size.

Preheat your oven to 475°F with a baking sheet or pizza stone inside. In a bowl, mix the low-moisture mozzarella and crumbled feta. Once the dough has risen, punch it down and divide it into two equal pieces. Roll each piece into an oval shape, fill the center with your creamy cheese blend, and pinch the edges to create a boat-like shape.

Place the shaped boats onto parchment-lined baking sheets, ensuring they have space between them. Brush the crust edges with an egg wash made from the remaining beaten egg. Bake for 13 to 15 minutes, or until the crust is golden brown and the cheese is bubbling.

Remove from the oven and carefully create a well in the melted cheese. Add a large egg yolk and a pat of cold unsalted butter in the center of each boat. Serve immediately.