In a large mixing bowl, whisk together the flour and salt. Make a well in the center of the mixture.

In a liquid measuring cup, whisk together 1⁄2 cup of water, the vegetable oil, and egg. Pour this into the well in the flour mixture and stir it all together until a shaggy dough forms.

Scrape the dough from the bowl onto a lightly floured surface and knead it together until smooth, about 6 minutes, sprinkling with additional flour as needed. (The dough should feel moist and soft, but once it starts to stick to your hands or the surface, add another sprinkle of flour.)

Wrap the dough with plastic wrap and let it rest at room temperature for 1 hour.

Meanwhile, peel and cut the potato into 1-inch pieces. Place them in a small pot and cover with cold water by 1 inch. Season the water with salt.

Bring the pot to a boil over medium-high heat, then reduce the heat to medium. Simmer the potatoes for about 12 minutes or until they are fork-tender. Drain the potatoes and return them to the pot.

Add the butter, sour cream, salt, and pepper to the pot. With a potato masher, mash the potatoes until they are very smooth. Place them in a large mixing bowl.

Stir in the cheese just until combined. Spread the mixture out on the bottom and around the inside of the bowl. Let it cool until it’s barely warm, about 45 minutes, stirring the mixture and spreading it occasionally. (You can speed this up by placing the bowl in the refrigerator, uncovered, once the potatoes are no longer steaming. )

Divide the dough into two pieces. Lightly flour a surface and a rolling pin. Working with one piece of dough at a time, roll the dough into a large piece, about ⅛-inch thick.

With a 3-inch round biscuit or cookie cutter, cut 10 to 12 circles from the dough. Set the dough circles on another floured surface (it’s ok if they overlap slightly).

Knead together the excess dough (from the first and second rolls) until it is a smooth ball. Wrap it in plastic wrap and let it rest for 15 minutes. Re-roll and cut this dough one time to make more circles.

Lightly flour a large, rimmed baking sheet and set it aside.

Fill each circle with about 1½ teaspoons of the potato filling mixture.

Brush the inside edges of the dough with water and fold them in half over the filling, pressing out any air in the interior. Press the edges all around to seal. (You can crimp the edges with a fork if you like.) As each pierogi is sealed, transfer it to the floured sheet tray in a single layer.

Fill a large pot two-thirds full with water and place over high heat and bring it to a boil.

Meanwhile, place 2 tablespoons of butter in a large skillet and heat it over medium-high heat. Once the butter is bubbling, add the onions and salt. Cook for 5 minutes, stirring frequently, until the onions start to soften and start to turn brown on the edges. Reduce the heat to medium and cook, stirring frequently, for about 10 minutes, until the onions are golden brown. (If the bottom of the pan gets too dark, add water, 1 tablespoon at a time, to deglaze.) Remove the onions from the heat.

Once the water is boiling, season it generously with salt. Working in batches, add 8 to 10 pierogi to the pot. Reduce the heat to medium to maintain a low boil and cook for 2 to 3 minutes until the pierogi float to the top of the pot and the noodle is tender but still chewy. Transfer with a slotted spoon or a spider to the skillet with the onions. Repeat with the remaining pierogi.

Return the skillet with onions and pierogi to medium heat. Cook for 3 to 4 minutes to rewarm, stirring gently to combine. (Add another tablespoon of butter if the pierogi stick to the skillet.) Season with salt and pepper to taste. Serve them immediately with some onions from the skillet and the sour cream.