Make the dill oil. The dill oil gets better as it sits, so make this first. In a small bowl, stir together the minced fresh dill, lemon zest, grated garlic, and olive oil. Stir in salt and pepper to taste and set aside.
Brown the sausage. Heat the oil over medium heat in a 5 qt dutch oven. Add the sliced kielbasa and brown it all over, stirring occasionally. This should take 7-8 minutes. Use a slotted spoon to transfer the sausage to a bowl.
Saute the cabbage. Increase the heat to medium-high. Once the oil and leftover fat in the dutch oven is very hot, add the cabbage and stir well to coat in the oil. Let the cabbage cook, stirring once or twice, until pieces are starting to brown and get soft. Add the onions and carrots and cook for another 2-3 minutes, until the onions are starting to soften.
Build the soup. Stir the paprika, ground caraway, and dijon mustard into the veggies. Pour in the chicken broth, add the sausage back in and bring to a gentle simmer. Season the soup with salt and lots of pepper, cover and simmer for 10 minutes.
Brown the pierogies. While the soup simmers, melt the butter in a nonstick skillet over medium heat. Once the butter is melted and foamy, add the pierogies to the pan in a single layer. Brown for 3-4 minutes, flip them, and brown for another 3-4 minutes. Remove from the heat and set aside.
Finish the soup. Remove the lid from the pot and add in the pierogies. Squeeze in the lemon, stir well, and season to taste with salt and pepper. Ladle the pierogi soup into bowl and drizzle with the dill oil.