Prepare the dough placing flour in a large bowl. Make a well in the center and pour in 1/3 cup olive oil, white wine, water, and salt. Mix together with your hands for a few minutes to form a smooth dough. Cover with plastic wrap and refrigerate while proceeding with recipe.
Heat olive oil in a large saucepan over medium heat. Stir in onion and garlic, and cook for a few minutes until the onion softens and turns translucent. Stir in the lamb and pork; increase heat to medium-high and continue cooking until the onion begins to brown, about 5 minutes.
Stir in tomato paste until the meat is covered. Stir in white wine, and simmer for 1 minute. Pour in water and bring to a simmer. Reduce heat to medium-low, cover, and simmer until tender, about 45 minutes.
When the meat is done, remove from the heat and allow to cool while preparing crust.
Preheat oven to 450 degrees F (230 degrees C). Lightly brush a 9x13-inch glass baking dish with olive oil.
Divide the dough into two pieces, with one piece larger than the other. Roll out the large piece on a floured surface until it is large enough to fit the bottom of the baking dish and come up the sides of the pan; press into the baking dish. Roll out the smaller piece of dough into a rectangle to use as the top crust; set aside.
Stir Kefalotiri cheese, potato, and rice into meat mixture. Season with parsley, mint, marjoram, salt, and pepper. Add egg and mix until well combined. Pour this mixture into the baking dish and top with remaining piece of dough. Brush the top with a little water and olive oil.
Bake pie in preheated oven for 1 hour until deep golden brown.