Flower Poğaça Rolls

To make the dough: In a large bowl, whisk together the milk, yeast, and sugar. Set aside until foamy, about 5 minutes., Whisk in the egg and oil. Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the flour and salt and mix to make a rough dough., Knead the dough for 8 minutes by hand or 6 minutes at medium speed in a stand mixer, until it forms a smooth ball., Place the dough in a greased bowl, cover, and let rise at room temperature until doubled in size, 1 to 1 1/2 hours., Turn the risen dough out onto a lightly floured surface. Divide it into 12 pieces. Roll each piece into a smooth ball. Cover with greased plastic and let rest for 30 minutes., To make the filling: Combine the feta and parsley in a small bowl. Set it aside., Line a baking sheet with parchment., To assemble: Roll each ball of dough into a 5" circle. Cut four slits from the edge toward the center, each about 1 1/2" long, starting at the four points of the compass. Place a generous tablespoon of the filling mixture in the center., Starting at the top, fold that quarter of the dough over the cheese to partially cover but not enclose it. Fold 1/4" of the dough's top edge back to form a petal shape. Fold the bottom quarter over the cheese and form a petal the same way, by folding back the top edge., Fold the right side over the cheese to make the petal shape, then the last quarter. For the last petal, tuck the edges underneath the roll, again folding the top edge back to make a petal., Place the shaped roll on the prepared baking sheet and repeat with the remaining balls of dough, leaving 2" between the rolls., Cover with greased plastic and let rise until doubled, 30 to 45 minutes. Toward the end of the rise time, preheat the oven to 350°F., To bake: Brush the rolls with the egg mixture. Bake until golden brown, 25 to 30 minutes. Remove from the oven and let cool on a rack for 10 minutes before serving warm., Store leftover rolls, well wrapped, in the refrigerator for up to five days.