Preheat the oven to 375 degrees F and line a baking sheet with parchment. Stir the pesto, mozzarella and 1/4 cup Parmesan together in a medium bowl. Cut the dough into 4 equal pieces. On a floured work surface, form the pieces into balls. Roll one ball into a 10-inch circle (about the size of a dinner plate) and place it on the prepared baking sheet. Spread with 1/3 of the filling, leaving a 1-inch border around the edges. Add 2 more layers of rolled dough circles and filling, ending with the fourth dough circle on top.
Mark the center of the dough by placing a small drinking glass (about 2 inches in diameter) in the middle. With a very sharp knife, make 4 cuts through the dough at 12, 3, 6 and 9 o'clock to make 4 quarters, leaving it intact at the center where the glass is. Cut through each quarter to make 8 equal-sized wedges, then cut through each wedge one more time to make 16 equal-sized wedges that are attached in the center. Lift a pair of 2 wedges and twist them away from each other, two turns in opposing directions. After twisting, place each pair back on the baking sheet. Repeat with the remaining pairs of wedges. Cover the dough with oiled plastic wrap and let it rise in a warm place until doubled in size, about 45 minutes.
Beat the egg and milk in a small bowl. Brush the egg wash all over the risen dough and sprinkle lightly with grated Parmesan and flaky sea salt if using. Bake until the bread is puffed and golden brown, 25 to 30 minutes. Cool on a wire rack 15 minutes before serving.
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