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Preheat oven to 425 degrees.
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Heat a large cast iron or other oven-safe skillet over medium heat. Add bacon and cook, stirring occasionally, until browning, about 7 minutes. Remove bacon from pan with a slotted spoon to a paper-towel lined plate.
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Season pork with salt and pepper. Turn heat to medium-high. Add pork tenderloins to the pan with the bacon fat and cook until browned, about 2 minutes per side. Turn with tongs to get all sides browned.
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Turn off heat and transfer the skillet to the oven. Roast for 10 to 15 minutes, until the internal temperature reaches 145 degrees. (While pork is roasting, continue to the next steps to make the sauce and kale.) Remove tenderloins to cutting board and loosely tent with aluminum foil. Let rest 10 minutes before slicing.
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Make the sauce: Melt 2 tablespoons butter in a skillet over medium heat. Add mushrooms and season with salt and pepper. Cook, stirring occasionally, until mushrooms release their juices, about 10 minutes. Add 1 tablespoon flour and cook for 1 more minute, stirring constantly. Add broth and cook for 5 to 10 more minutes, until slightly thickened. Remove from heat and let cool for a few minutes. Right before you eat, mix in sour cream.
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Make the kale: Heat olive oil over medium heat in a pot, saucepan, or large skillet (really whatever you have left at this point). Add kale and season with salt and pepper. Cook, stirring occasionally, for 5 minutes. Stir in chopped bacon and cook 2 more minutes.
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To serve: Divide dumplings, sliced pork tenderloin, and kale between serving plates. Spoon the mushroom sauce over pork and dumplings, or serve on the side. Garnish with dill.