Preparation

Make the dough

• In the bowl of a stand mixer, combine the milk, yeast, and sugar. Set aside until foamy, about 5 minutes. Whisk in the egg and oil. Put the dough hook on the stand mixer. Add the flour and salt, and mix on low speed until the flour is incorporated, scraping the bowl as needed. Increase the speed to medium low, and knead until smooth and stretchy, 8 to 10 minutes. (Alternatively, combine and knead the dry ingredients by hand.)
• Lightly flour a work surface, and lightly oil a large bowl. turn the dough out onto the surface and sprinkle lightly with flour. Shape into a ball, and transfer to the bowl. Cover with plastic wrap and let rise at room temperature until doubled in size, 1 to 1-1/2 hours.

Make the filling

• In a medium bowl, combine the feta and parsley.

To shape and bake oval rolls

• Lightly flour a work surface, and turn the dough onto it. Deflate the dough and divide into 17 pieces, about 1-1/2 oz. each. Roll each piece into a ball, cover with plastic wrap, and let rise at room temperature until an indentation remains after you press a roll with a finger, 30 to 40 minutes.
• Line two large rimmed baking sheets with parchment.
• Press or roll a dough ball into a 4- to 4-1/2-inch circle. Place about 1-1/2 Tbs. of the filling in the center of the dough circle. Pull up two sides to enclose the filling. Seal the dough by pinching with your fingertips. Turn the roll over so that it’s seam side down, and gently roll into an oval or football shape. Place seam side down on the prepared baking sheet, and cover with plastic wrap. Repeat with the remaining balls of dough, leaving at least 2 inches between the rolls. Let rise at room temperature until almost doubled in size, 30 to 40 minutes.
• Position a rack in the center of the oven and heat to 350°F. Whisk the egg and milk, and lightly brush each roll with the wash. Bake, swapping and rotating the pans about halfway through, until light golden brown, 28 to 33 minutes. cool on the baking sheet set on a wire rack. Serve warm or at room temperature, preferably on the same day.

To shape and bake rose-shaped rolls

• Lightly flour a work surface, and turn the dough onto it. Deflate the dough, and divide into 13 pieces, about 2 oz. each. Roll each piece into a ball, cover with plastic wrap, and let rise at room temperature until an indentation remains after you press a roll with a finger, 30 to 40 minutes.
• Line two large rimmed baking sheets with parchment. Lightly flour a work surface.
• Roll or press a dough ball into a 5-inch circle. Starting from the edge and heading toward the center, cut four evenly spaced slits, each about 1-1/2 inches long, into the circle.
• Place about 1-1/2 Tbs. of the filling in the center. Wrap one quarter of the dough around the filling; the dough will partially cover but not fully enclose the filling. As you wrap, fold ¼ inch of the top of the dough edge back to form an open petal. Fold the dough quarter opposite the first quarter around the filling and fold back the top to form an open petal. Secure the dough on the bottom with a pinch. Wrap and fold a third quarter in the same manner. Then, when folding the final quarter over, tuck the edges underneath the rose, and pinch to secure. Place on the prepared baking sheet, and cover with plastic wrap. Repeat with the remaining balls of dough, leaving at least 2 inches between the rolls. Let rise at room temperature, until almost doubled, 30 to 45 minutes.
• Position a rack in the center of the oven and heat the oven to 350°F. Whisk the egg and milk. Lightly brush the dough with the egg wash.
• Bake, swapping and rotating the pans about halfway through, until golden brown, 25 to 30 minutes. Cool on the baking sheet set on a wire rack. Serve warm or room temperature, preferably on the same day.