Mix the flour and salt in the large bowl. Cut the lard and butter into small chunks. Cut in the chunks into the flour mixture until it becomes crumble. Beat the 1egg with the vinegar and water. Stir in the egg mixture slowly until mixture form a ball. Wrap in plastic and put in the refrigerator for at least 1 hour.
Preheat the oven 350°F. Peel and chop the onion, carrots, celery, and mushrooms into 1/4-inch sized-dice. Place vegetables into a large frying pan on a medium heat with 2 tbsp of olive oil. Pick the rosemary and thyme leaves off the woody stalks, finely chop them, and add to the pan. Add a pinch of salt and pepper. Fry and stir for 10 minutes. Put the vegetable mixture into a large bowl.
Soak the porchini in 1/2 cup of water. Peel and chop the shallot and garlic. Finely slice the bacon. Place shallot, garlic, and bacon into a large frying pan on a high heat with 1 tbsp of olive oil. Fly and stir for a few minutes, add the ground beef, a pinch of salt and pepper. When the ground beef becomes brown, add the tomato, beef broth and beef base. Lower the heat, simmer for 30 minutes until almost a little bit liquid. Add salt and pepper to taste. Add the vegetable mixture and chopped porchini into the beef mixture and mix well. Add the flour, and turn the stove off.
Peel the potatoes, put them into a pan of water. Boil them for about 10 minutes about tender. Drain in a colander and return them to the pan. Add the butter and milk, and mash until smooth. Add the chopped parsley and mix well.
Put the mashed potato in the 9.5 inch baking dish. Put the filling over the potato. Sprinkle the diced parmigiano on the filling. Roll out the dough into a 10 inch round. Place the crust over the filling, press the edge down on the baking dish to seal using fork. Make a small hole in the center of the crust and 4 to 6 slits in the crust. Brush the crust with beaten egg. Bake for 45 minutes.