Preheat the oven to 400°F.
In a pot of boiling water, cook the pierogies according to the directions on the package. Drain and set aside.
In a sauté pan over medium heat, add the cabbage, vinegar, broth, and season with salt and pepper. Cover with a lid and bring to a simmer for about 20 minutes. Stirring occasionally until the cabbage is tender. Adjust the seasoning with salt as needed to taste.
Place the kielbasa on a foil-lined baking sheet and cook on both sides until nicely browned, about 7-8 minutes per sides to an internal temperature of 155°F using a thermometer.
In a non-stick pan over medium heat, melt the butter. Once the butter starts to sizzle, brown the pierogies on both sides until they are golden brown.
Serve the kielbasa with cabbage and pierogies. Serve with sour cream and mustard (if desired) with crusty bread on the side.