For the Dough:
2 cups unbleached all-purpose flour, plus extra for the work surface
½ teaspoon salt
1 egg yolk
1 tablespoon vegetable oil
4 to 6 tablespoons cold water
For the Caramelized Onions:
4 tablespoons unsalted butter
3 large onions, halved and thinly sliced
¼ teaspoon salt
For the Potato and Cheese Filling:
1 medium russet potato (about 9 ounces), peeled and sliced 3/4 inch thick
1 tablespoon salt
¼ cup reserved chopped caramelized onions
1½ ounces cheddar cheese, shredded (about ⅓ cup)
1½ ounces farmers cheese, cut into small pieces (about ¼ cup)
1 tablespoon unsalted butter
Salt and ground black pepper, to taste
For Cooking and Serving:
1 tablespoon salt
4 tablespoons unsalted butter, for sautéing
Sour cream, for serving
Make the Pierogi Dough: Mix the flour an salt in a small bowl. Pour out onto a clean, flat surface and hollow out the middle. Add the whole egg, egg yolk, and oil in the middle and slowly mix to combine. Slowly add 4 tablespoons of the water until the dough forms a ball. If the dough doesn’t ball up, add the remaining water, 1 tablespoon at a time. Transfer the dough to a lightly floured work surface and knead by hand until it firms slightly and becomes smooth, about 2 minutes. Cover with plastic wrap and set aside to relax for at least 15 minutes or up to 2 hours.
Make the Caramelized Onions: Meanwhile, melt the butter in a 12-inch nonstick skillet over medium heat. Add the onions and salt and cook until very soft and well browned, 15 to 20 minutes. Measure out ¼ cup of the onions, chop them fine, and reserve them for the filling. Cover the caramelized onions to keep warm.
Make the Potato and Cheese Filling: Cover the potatoes by 1 inch of water in a large saucepan and add 1 tablespoon salt. Bring to a boil, then reduce to a simmer and cook until the potatoes are tender and a fork can be slipped easily into the center, 10 to 12 minutes. Drain the potatoes into a colander.
Set a food mill or ricer over a medium bowl and process the potatoes into the bowl. Add the caramelized onions. Stir in the cheeses and butter until incorporated and season with salt and pepper to taste. Cool slightly before filling the pierogi or cover with plastic wrap and refrigerate until needed, up to 2 days.
Assemble the Pierogi: Dust a baking sheet liberally with flour; set aside. Divide the dough into 2 even pieces and cover with plastic wrap. Working with 1 piece of dough at a time, unwrap the dough and roll out on a lightly floured work surface into a 15-inch circle, about 1/16-inch thick. Using a 3-inch round biscuit cutter, cut out as many rounds as possible. Carefully gather up the dough scraps, wrap them in plastic wrap, and set aside.
Place 1 teaspoon of the filling in the center of each dough round and moisten the edge of the dough round with water, using either your finger or a pastry brush. Fold the dough in half over the filling, making a half-moon shape. Pinch the seam along the edge to secure. Crimp the seam using the tines of a fork. Transfer to the prepared baking sheet and cover with a damp kitchen towel. Repeat with the remaining dough.
Cook the Pierogi: Method 1: Bring 4 quarts of water to a boil in a large pot. Add 1 tablespoon of salt and half the pierogi. Cook, stirring often, until the edges feel al dente, 5 to 6 minutes (8 to 10 minutes if frozen). Using a wire spider or slotted spoon, transfer the pierogi to a colander and set aside. Return the water to a boil and cook the remaining pierogi. Method 2: line a baking sheet with parchment paper and bake in a preheated 350deg oven for 20 mins or until crisp on the outsi
While the second batch of pierogi is boiling, melt 2 tablespoons of the butter in a 12-inch nonstick skillet over medium-high heat. Add the first batch of boiled and drained pierogi and sauté until golden on both sides, 1 to 2 minutes per side. Transfer the browned pierogi to a platter and cover to keep warm. Drain and sauté the remaining pierogi using the remaining 2 tablespoons butter. Sprinkle the caramelized onions over the top and serve with sour cream.