For the dough, mix together the flour, onion powder, and salt in a large bowl until combined. Cut the butter into pea-size pieces over the bowl of flour. Pinch the butter with the flour between your fingers until a crumbly sand-like mixture forms. Add the egg and knead it into the dough. Add the vegetable stock, 1 tablespoon at a time, kneading until a smooth dough forms. Cover and refrigerate for at least 4 hours.
For the filling, heat the olive oil in a large frying pan over medium heat. Add the onions, garlic, and bay leaf and sauté until the onions have softened, about 5 minutes. Add the regular potatoes, sweet potatoes, and beet and stir to combine. Allow to cook for about 10 minutes, stirring about every 3 minutes. Add the chicken stock, salt, pepper, oregano, mustard powder, and garlic powder and stir to combine. Allow the mixture to continue to cook until the liquid has evaporated and the beets have nearly (but not completely) cooked through, about 10 to 15 minutes. Remove from heat and set aside to cool to a warm temperature. Stir in the feta.
Preheat the oven to 375 degrees Fahrenheit. Roll out the dough on a clean and lightly floured work surface until it is about 1/8-inch thick. Use a 4-inch circular cookie cutter to cut out one dough round. Spoon about 1 heaping teaspoon of the filling onto the bottom half of the round. Dip your finger in a glass of water and trace it around the edge of the top half of the dough. Fold it over the bottom half and press the edges together to seal.
To create the the braided-looking edge start at one corner of the sealed empanada and fold the corner up over the top of the empanada, pinching it together with the top of the empanada once you fold it. Repeat this again with the new corner that formed at the base of the last fold and continue until sealed all the way along the edge. You can watch the technique here. Place the empanada on a baking sheet.
Repeat with the remaining ingredients, spacing the empanadas about 2 inches apart on the baking sheet. This may take a while, so I recommend keeping a kitchen towel over the rolled-out dough to keep it from drying out and becoming unpliable. Once you’ve filled the sheet with empanadas, brush the empanadas with the egg wash and bake for 25 to 30 minutes, until puffy and golden. Let cool for 30 minutes before serving.