Season the pork generously with kosher salt and black pepper. Heat a large Dutch Oven over medium heat, and in batches, cook in the oil, searing the pieces on all sides. Take your time with this step, usually in 2 batches, reserving the seared pieces to a plate on the side as you go. You are building flavor here, in total about 20 minutes.
If you have a lot of fat in the pot, pour some out - you want a thin layer to start cooking the onions. With the heat still at medium, cook the onions, stirring for 5-7 minutes, allowing them to break down and start to color a bit - then add the garlic, stirring for another 2 minutes or so. Season with salt and pepper. Add the tomato paste, paprika and crushed red pepper flakes, cooking for another 2 minutes.
Add the vinegar, 4 cups stock, 4 cups water, the reserved pork and any juices that remain in the plate. Season with salt and pepper, and bring to a simmer. Reduce the heat until just barely bubbling, and cook over this low heat for at least 3 hours. You will likely need to add water or stock to the pot as you go - I found I needed 2 extra cups of water. When the pork is fork tender (falling apart tender) your soup is on its way! If you need more time for this, just keep going. The broth should taste very good (season again with salt and pepper if not) and the cabbage should be almost melty in texture.
At this point, add the potatoes, kale, kielbasa and juice from 1/2 a lemon. Simmer for another 10-15 minutes, until the potatoes are fully cooked through and the kale wilted. Taste for seasoning one more time, and you are good to go!
Finish the soup by serving with thinly sliced lemon and chopped dill fronds.