Add the diced onions with the butter to a pot/pan with a good fitting lid. Caramelize the onions over medium to low heat until browned. Stir often. According to Baba, the tight cover keeps the onions tender. Refrigerate until ready to serve.
Beat the eggs, milk, onion butter, water, sugar, and salt together in a bowl. Add the flour, 1 cup at a time, until fully blended.
Turn out dough onto a lightly floured surface and knead until smooth. Allow dough to rest under an overturned bowl for 20 minutes.
Mash the boiled potatoes and stir in the cheeses until combined.
Dust a parchment-lined baking sheet with flour and set aside.
Pinch off a half-walnut-sized piece of dough and, using a rolling pin, roll it into a very thin disk.
Place an approximate tablespoon of filling into the center of the dough circle. Moisten the edges of the dough with water then then fold the dough in half, to cover the filling. Press the edges together, with floured hands, to seal.
Place the pirohi on the prepared baking sheet. Repeat with the remaining dough and filling.
Fill a Dutch oven with water and bring to a boil. Add 5 pirohi to the boiling water and cook for 3 minutes or until they float to the surface. Removed the cooked pirohi with a slotted spoon and place on a wire rack coated with cooking spray. Repeat procedure with remaining pirohi.
Melt 2 tablespoons butter in a large nonstick skillet over medium heat. Add 12 pirohi and saute 5 minutes, flipping over half way through, until warmed through and lightly browned. Sprinkle some of the caramelized onions over the pirohi and stir until they are coated and the onions are warmed. Remove to a serving platter and keep warm while you repeat with the remaining pirohi.
If you wish to freeze extra pirohi, prior to boiling them, place them in a single layer on a baking sheet or plate. Cover with plastic wrap and freeze until solid, at least 2 - 3 hours. Transfer them to a freezer-safe ziplock bag and store in the freezer for up to 3 months.
When ready to cook the frozen pirohi, take them directly from the freezer and drop into boiling water. Once cooked through, proceed with the remaining directions.
Baba would spend from dawn until late night making her pirohi for her family, and froze a lot of it for future dinners. I scaled down the recipe as follows... and it still made 80 pirohi!
For the dough: 2-1/2 cups flour + more for kneading, 1 egg, 1/4 cup milk, 1/2 cup potato water, 2 tablespoons butter from onion mixture, 1/2 teaspoon salt, pinch of sugar.
For the filling: 1-3/4 pounds potatoes, 4 ounces farmer's cheese, 4 ounces cream cheese, 3 slices American cheese.
Topping: I made the same amount as listed in Baba's original recipe since I surmise she made the caramelized onions fresh every time she used the frozen pirohi. I cooked half the pirohi and froze the remaining and the caramelized onions were just about the right amount.
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