To make the pierogi dough: Mix the 4 eggs, butter, sour cream, and salt together in a bowl with a whisk. Sift the flour onto a flat work surface and make a well in the center, pour the wet ingredients into the crater. Stir the mixture together with a fork, gradually incorporating the flour into the well until a soft dough forms. Squeeze the dough with your hands, if it¿s too sticky, add a bit of flour; if it¿s not pliable, add a couple of drops of water. Knead the dough, adding only as much additional flour as needed to keep the dough from sticking, until it is smooth and elastic, about 5 minutes. Gather the dough into a ball and cover it with an inverted bowl or plastic wrap for 10 minutes or so to rest.
To make the filling: Cook the potato cubes in boiling salted water until tender, about 15 minutes. Drain and then mash them with a ricer or regular old potato masher while they are still hot; set them aside to cool while preparing the rest of the filling. Trim the root off the end of the leek and cut off all but 1-inch of the green part. Halve the leek lengthwise and rinse really well under cool water, checking for dirt everywhere; it gets trapped in all the layers of outer leaves. Slice the leek finely. Melt 2 tablespoons of butter in a large skillet over medium-low heat. Add the leek, mushrooms, and thyme, saute for a few minutes to soften the vegetables; season with salt and pepper. Continue to cook until the vegetables are broken down and almost dry, then pour in the cream, stir to incorporate. Remove the pan from the heat, take out the sprigs of thyme, and scrape the mushroom mixture into the potatoes, mix well to incorporate the ingredients. Make sure the filling is not too hot, or it will start to cook the dough, causing it to sog.
Lightly flour your rolling pin and counter. Take 1/2 of the dough (leaving the rest covered so it doesn¿t dry out while you work) and roll it out into a thin circle, about 1/8-inch thick. Using a 3-inch round cookie or biscuit cutter, cut circles out of the dough. Spoon 1 tablespoon of the filling into the center of each pastry circle, leaving a 1/2-inch border. Beat the remaining egg with 2 tablespoons of water to make an egg wash and brush it on the edges of the circle. Fold the dough over in 1/2 to enclose the filling and form a semi-circle and seal the edges by crimping with the tines of a fork. Lightly tap the bottoms of the dumplings on the counter to make it flat. Repeat with the remaining half of dough and filling. You should make about 35 pierogies; put 1/2 of them in the freezer to keep on hand (they are better than the ones you find in the frozen section of the grocery store.)
Coat a large saute pan (that has a tight fitting lid) with vegetable oil and place over medium heat. Lay the pierogies in the pan (you may have to do this in batches) and fry them for a couple of minutes until they start to crisp and brown on both sides. Pour in 1 cup of water and cover the pan to let the pierogies steam; when the water evaporates the turnovers should be cooked and crisp. Arrange the pierogies on a platter, garnish with parsley, and serve with roasted apples and/or sour cream.
Preheat the oven to 400 degrees F. Spread the apples out in a casserole pan and add the lemon juice, cinnamon, sugar, salt, and butter. Bake until the apples are roasted and soft, about 30 minutes. Serve the chunky apple sauce with pierogies, either hot or chilled.