To prepare the dough, place the flour, salt and baking powder into a large bowl. Add the melted butter and hot water, and mix using a spoon. When mixed, it will still feel a bit dry, add the egg and mix. If you add the egg with the hot butter, it can cook the egg, which ruins the dough.
Place the dough on a lightly floured surface. The dough will be soft, so gently knead to keep the dough soft.
Prepare the vegetables. Mix together the ground beef, thinly sliced vegetables, black pepper, salt, water and oil.
Cut the dough in ⅔ and ⅓ pieces. The ⅔ piece will be used as the base of the Pirog pie, and the ⅓ piece will be the cover. Roll the dough into about ½cm or ¼" thick.
Put the larger dough inside a 8.5" x 11.5 baking pan, including up the sides. You can lightly grease the pan if you like, however the dough has enough butter so I find it doesn't stick. Place the meat mixture inside and cover with the small piece of dough. Pinch the dough edges together to seal. Cut off any excess dough and set aside to make a dough pattern on top.
Using a fork, pierce the dough randomly across the top, to allow steam to escape when cooking. Ensure the holes to seal up, you can also cut a round circle in the middle. Roll out the excess dough, and cut some leaf patterns. Place on top of the pie. Beat the egg yolk and brush the dough for a golden finish.
Preheat oven to 350°F/180°C. Place into the oven and cook for about 50-60 minutes until golden brown, the potatoes are soft and the meat runs clear.