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Pat meat dry with paper towels. Sprinkle with salt and pepper. Heat oil in heavy large skillet over high heat until very hot. Working in batches, add meat in single layer and cook just until brown on outside, about 1 minute per side. Transfer to rimmed baking sheet.
Melt 2 tablespoons butter in same skillet over medium-high heat. Add chopped shallots and sauté until tender, scraping up browned bits, about 2 minutes. Add button mushrooms. Sprinkle with pepper and sauté until liquid evaporates, about 12 minutes. Add beef broth, then Cognac. Simmer until liquid thickens and just coats mushrooms, about 14 minutes. Stir in crème fraîche and Dijon mustard. Add meat and any accumulated juices from baking sheet. Simmer over medium-low heat until meat is heated through but still medium-rare, about 2 minutes. Stir in chopped dill. Season to taste with salt and pepper.
Meanwhile, cook noodles in large pot of boiling salted water until tender, about 8 minutes. Drain. Transfer to bowl. Add remaining 4 tablespoons butter and toss to coat. Season with salt and pepper. Divide noodles among plates. Top with beef and sauce. Sprinkle generously with paprika.