Bring a large pot of salted water to a boil. Add the egg noodles and cook as the label directs. Reserve 1/2 cup cooking water, then drain.
Meanwhile, heat the olive oil in a large pot or Dutch oven over medium-high heat. Add the sausage and cook, breaking it up with a wooden spoon, until crumbled and lightly browned, about 3 minutes. Add the onion; cook, stirring occasionally, until softened, about 3 minutes. Add the garlic, red peppers and paprika and cook, stirring, 1 minute. Add the chicken broth, cabbage and 1/2 teaspoon salt. Bring to a gentle simmer and cook until the cabbage is tender, about 4 minutes.
Add the noodles and sour cream to the sausage mixture. Cook, stirring, until well coated, about 30 seconds (do not boil). Add the reserved cooking water as needed to loosen. Stir in the dill and season with salt and pepper.