In the bowl of a stand mixer, combine 4 cups all-purpose flour and 2 teaspoons salt. Add ½ cup vegetable oil, 2 large eggs, and ¾ cup warm water. Fit the mixer with the dough hook and mix on low until the ingredients are combined.
Increase the mixer speed to medium-high and knead the dough with the dough hook for 5–6 minutes, until the dough is smooth and springy.
Lightly oil a bowl with a little of the vegetable oil, transfer the dough to the oiled bowl, cover, and let rest for 30 minutes.
While the dough rests, prepare the filling. Peel and cut 2 medium russet potatoes into large pieces. Place the potatoes in a pot, add enough cold water to cover them by about 1 inch, and bring to a boil over high heat.
Boil the potatoes 20–25 minutes, until fork-tender. Drain the potatoes and let them cool 5–10 minutes, then mash until smooth.
Stir 2 tablespoons unsalted butter, ¼ cup cottage cheese or sour cream, 1 cup shredded cheddar cheese, and 1 teaspoon salt into the mashed potatoes until the filling is smooth and evenly combined. Set the filling aside.
Divide the rested dough into two equal pieces. Keep one piece covered to prevent drying while you work with the other.
On a lightly floured surface, roll one piece of dough to about 1/8-inch thickness. Use a 3-inch cutter to cut circles as close together as possible; gather scraps, re-roll, and cut more circles.
Place about ½ tablespoon of the potato filling in the center of each dough circle. Fold each circle in half over the filling and press the edges together firmly to seal. Ensure the edges are well sealed so the filling does not leak. Set finished pierogis aside on a lightly floured surface and cover loosely to prevent drying.
Repeat steps 8–9 with the remaining dough and filling.
Slice 1 small sweet onion. In a large skillet, melt 3 tablespoons unsalted butter over medium-low heat. Add the sliced onion and cook, stirring occasionally, about 20 minutes, until the onions are nicely browned. Keep the skillet on low heat after the onions are done.
Bring a large pot of water to a rolling boil over high heat. Working in two batches, gently add the pierogis to the boiling water and cook until they float, about 3 minutes per batch.
Using a slotted spoon, transfer the cooked pierogis directly from the boiling water into the skillet with the browned onions. Cook the pierogis in the skillet over medium heat 2–3 minutes to brown and finish heating.
Serve the pierogis hot with the browned onions.