1 tablespoons unsalted butter – melted (for brushing after baking)
Instructions
In a large bowl, combine the warmed milk and sugar, then sprinkle the yeast over the top. Let stand until foamy, 5–10 minutes.
Add the flour and salt to the bowl, then mix in the neutral oil. Stir until a shaggy dough forms, then knead on a lightly floured surface until smooth and elastic, 8–10 minutes.
Place the dough in a lightly oiled bowl, cover, and let rise in a warm place until doubled in size, 60–75 minutes.
Meanwhile, make the filling: combine the grated mozzarella, crumbled feta, and softened butter in a bowl and mix into a cohesive cheese mixture.
Line a large baking sheet with parchment. Preheat the oven to 475°F (245°C) with a rack in the center.
Turn the risen dough out and divide into 2 equal pieces. Shape each into a ball, cover, and rest for 10 minutes.
Working with one piece at a time, roll into a 10–11 inch circle. Mound half of the cheese mixture in the center. Lift the edges up over the filling to enclose it, pinch to seal, flip seam-side down, and gently pat or roll into an even 8–9 inch round. Repeat with the second piece.
Transfer both rounds to the prepared sheet. Cut a small steam vent in the center of each. Whisk the beaten egg with 1 tablespoon milk and brush the tops.
Bake until deep golden with browned spots and the cheese is melted and bubbling at the vent, 12–16 minutes.
Brush the hot breads with melted butter. Cool 5 minutes, then cut into wedges and serve warm.