Preheat your oven to 450°F. Toss the sweet potatoes and red onion slices in the oil and seasonings and lay on a thin layer on a roasting pan. Roast for 20 minutes, stirring halfway through. Allow to cool before adding to the pastry. Turn the oven down to 400°F when finished.
While the potatoes and onions roast, prepare your dough. Roll out and cut into 9 squares. Turn the squares so that they are diamonds, and use the tip of a sharp knife to cut about 1/4 inch in from the outside of the dough, starting at the top point and ending at the bottom (see photo in post for reference). Don't fully cut through the top or bottom points. Place the diamonds back in the fridge to chill while you wait for your potatoes and onions to cool.
When everything is cool, place potatoes and onions in the middle of each diamond. Take the left and right points of the diamond and cross them over each other so that they form a ridge of dough around the filling. Remember, we left the top and bottom points connected? There should be a twist there (see photos in post for reference). Add feta cheese to each kite (tucking it around the filling so it doesn't burn when exposed to heat) and brush the filling with balsamic vinegar. Then brush the pastry with beaten egg before placing in the oven.
Bake at 400°F for 20-25 minutes, until pastry is golden brown. Allow to cool a little before serving, but serve warm while the pastry is crisp. These are best enjoyed immediately, but will keep wrapped on the countertop for up to two days. To reheat, warm in an oven (microwaves and pastry do not mix). Enjoy!