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1
Preheat oven to 350u00b0F. Toast walnuts on a rimmed baking sheet, tossing once, until golden brown, 7-10 minutes. Let cool, then chop.
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2
Heat 3 Tbsp. oil in a large skillet over medium-high. Season pork loin (look for one with a nice dark color and some fat on it) with salt and cook until browned on all sides, 10-15 minutes. Transfer pork to a platter.
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3
Add onions and beets to same skillet and cook, stirring often, until onions are slightly translucent and browned, 10-12 minutes; season with salt and pepper. Add garlic, sauerkraut, brine, broth, wine, and 2 Tbsp. lemon juice and bring to a simmer. Cover skillet, reduce heat to low, and cook until beets are fork-tender, 25-30 minutes.
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4
Return pork to skillet and push down into liquid. Turn heat up to medium and simmer, uncovered, turning meat occasionally until an instant-read thermometer inserted into the thickest part of loin registers 150u00b0F, 20-30 minutes. Transfer pork to a cutting board and let rest 5 minutes before slicing.
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5
Meanwhile, mix walnuts, mint, parsley, apricots, remaining 2 Tbsp. lemon juice, and remaining 1/2 cup oil in a small bowl; season with salt and pepper.
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6
Spoon beet mixture onto plates and arrange sliced pork on top. Spoon a generous amount of walnut sauce over.
