Toss the beef with the paprika, 1 1/2 teaspoons salt and 1/2 teaspoon pepper. Coat in flour and shake off any excess. Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high heat. Add the beef and cook undisturbed until it begins to brown, about 3 minutes. Continue to cook, turning the beef as needed, until mostly browned, about 3 minutes more. Remove the skillet from the heat and transfer the beef to the insert of a 6-quart slow cooker; add the potatoes, mushrooms, carrots and onions and stir to combine.
Heat the remaining 1 tablespoon of oil in the skillet over medium heat. Add the tomato paste and stir until the oil begins to turn brick-red, about 1 minute. Add the flour and wine and whisk until thick (it's OK if there are some lumps). Add the beef broth, thyme, caraway if using, 1/2 teaspoon salt and a few grinds of pepper and bring to a simmer, whisking; continue simmering and whisking until the gravy is smooth and thick, about 4 minutes.
Pour the gravy into the slow cooker, cover and cook on low for 8 hours. The meat and vegetables should be tender.
Season with salt and pepper and stir in the parsley. Serve the stew in bowls with dollops of sour cream and a sprinkle of paprika.
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