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Pat the beef dry with a paper towel; sprinkle with the pepper. Heat 1 tablespoon of the oil in a large nonstick skillet over medium-high. Add the beef to the skillet; cook, stirring occasionally, until lightly browned on all sides, about 4 minutes. Transfer the beef to a 5- to 6-quart slow cooker. Add the carrots, onions, and remaining 1 tablespoon oil to the skillet; cook, stirring occasionally, until the vegetables are softened, 3 to 4 minutes. Transfer to the slow cooker.
Add the wine to the skillet; cook over medium-high 1 minute, stirring and scraping to loosen the browned bits from the bottom of the skillet. Pour over the beef mixture in the slow cooker.
Stir the stock, tomatoes, and pesto into the slow cooker. Cover; cook on HIGH 2 hours. Stir in the barley; cover and cook on HIGH until the beef and barley are tender, about 2 hours. Stir in the kale and salt. Ladle the soup into bowls, and serve hot.