Preheat an oven to 375°F Toss onions with 1 tablespoon oil on a large rimmed baking sheet. Roast until lightly browned, about 45 minutes, tossing once halfway through. Remove from oven and pour 1/2 cup beef broth over the onions, scraping up any browned bits from the pan. Scrape into a slow cooker.
While onions roast, mix the flour, caraway seed, 1 tablespoon paprika, and salt in a bowl. Add beef and toss to coat. Pat off excess flour and reserve remaining flour mixture.
Heat 1 tablespoon oil in a large heavy skillet over medium high heat and brown beef on all sides in 2 batches. Add the remaining tablespoon oil for the second batch. Transfer the browned meat to the slow cooker.
Add the reserved flour to the pan, and cook, stirring, until lightly browned, about 4 minutes. Stir in remaining 1/2 c beef broth, the brown sugar, ketchup, vinegar, and 1 tablespoon of the remaining paprika. Bring to a boil and pour in slow cooker. Cover and cook 3 to 4 hours on high, or 6 to 8 hours on low, until onions are practically liquefied and beef is tender.
Stir in remaining paprika and the parsley and cook for 5 minutes. Serve over cooked egg noodles if desired.