Preheat the oven to 400°F. Line a large rimmed baking sheet with parchment paper or a silicone mat.
While waiting for the oven to heat up, melt butter in a medium skillet over medium heat. Add the strips of onion and a generous pinch each of salt and pepper. Cook, stirring occasionally, for about 10 minutes.
2 Tablespoons butter, 1 medium onion, kosher salt and black pepper
While the onions get started, cut the Brussels sprouts into halves (cut any really large ones again into quarters). Toss in a bowl with a light drizzle of olive oil and sprinkle of salt and pepper. Spread Brussels sprouts out on one half of the sheet pan and place in the oven.
10-12 ounces Brussels sprouts, extra-virgin olive oil
Place the frozen pierogies in the same bowl that the Brussels sprouts were in. Add another pinch of salt and pepper, 1 teaspoon apple cider vinegar, and the mustard. Use a rubber spatula to stir and add just enough olive oil to lightly coat all the pierogies.
16 ounces frozen pierogies, 1 Tablespoon plus 1 teaspoon apple cider vinegar, 1 teaspoon whole grain mustard
Arrange pierogies on the other half of the baking sheet, minimizing any overlap. Do another quick check—if any parts of the pierogies look bare, brush a tiny bit more olive oil on top.
Transfer to the oven and roast for another 20-25 minutes, pausing around the halfway mark to toss the Brussels sprouts and flip over the pierogies.
Once the pierogies are in the oven, return to the onions. Stir in 1 Tablespoon apple cider vinegar and continue to cook over medium heat. They should develop a sweet aroma and turn dark brown—ie., caramelize. Once they become as brown as you like, turn off the heat and leave onions in the skillet.
When the pierogies and Brussels sprouts are fork-tender, remove from the oven and top generously with the caramelized onions. Serve right away with more salt, pepper, and mustard for dipping, if desired.