Heat 2 teaspoons oil in a large skillet over medium heat. Add half the beef and brown on all sides, turning frequently, about 5 minutes. Transfer to a 6-quart slow cooker. Drain any fat from the pan. Add the remaining 2 teaspoons oil and brown the remaining beef. Transfer to the slow cooker.
Return the skillet to medium heat, add mushrooms and cook, stirring often, until they give off their liquid and it evaporates to a glaze, 5 to 7 minutes. Sprinkle flour over the mushrooms; cook undisturbed for 10 seconds, then stir and cook for 30 seconds more. Pour in ale (or beer); bring to a boil, whisking constantly to reduce foaming, until thickened and bubbling, about 3 minutes. Transfer the mushroom mixture to the slow cooker.
Add carrots, onion, garlic, mustard, caraway seeds, salt, pepper and bay leaf to the slow cooker. Stir to combine.
Put the lid on and cook on low until the beef is very tender, about 8 hours. Discard the bay leaf before serving.
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