In the work bowl of a food processor, pulse radishes until finely minced. Add goat cheese, rosemary, parsley, salt, and pepper. Pulse until combined and smooth. Spoon mixture into a bowl, cover, and refrigerate for 30 minutes to 1 hour.
Preheat oven to 400° and line 2 baking sheets with parchment paper.
On a lightly floured surface, unroll puff pastry sheets. Using a 1¾-inch fluted cutter, cut 16 rounds from each pastry sheet. Using a 1-inch round cutter, cut 1-inch rounds from the centers of 16 rounds, reserving centers for another use. Place a pastry ring on top of each whole round, using water to adhere. Place pastries at least 1 inch apart on prepared pans. Refrigerate until firm, at least 20 minutes.
In a small bowl, whisk together egg and 1 tablespoon water. Brush pastries with egg wash.
Bake pastries until puffed and golden brown, 8 to 10 minutes. Let cool on pans for 5 minutes, and move to a wire rack to let cool completely.
Spoon chilled radish mixture into a piping bag and cut a ¼-inch opening. Fill each pastry evenly with filling, approximately ½ tablespoon each. Garnish with rosemary, if desired. Serve immediately.
Recipe by Victoria at https://victoriamag.com/radish-and-goat-cheese-vol-au-vents/