1/4 cup walnuts
3 tablespoons plus 1/2 cup olive oil
1 1/2 pounds pork tenderloin
2 medium onions, coarsely chopped
3 medium golden beets, scrubbed, cut into bite-size pieces
Freshly ground black pepper
4 garlic cloves, chopped
2 cups sauerkraut, plus 1/2 cup brine
1 cup low-sodium chicken broth
1/2 cup dry white wine
4 tablespoons fresh lemon juice, divided
1/2 cup finely chopped mint
1/2 cup finely chopped parsley
1/4 cup finely chopped dried tart apricots
Preheat oven to 350°F. Toast walnuts on a rimmed baking sheet, tossing once, until golden brown, 7–10 minutes. Let cool, then chop.
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.
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.
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 150°F, 20–30 minutes. Transfer pork to a cutting board and let rest 5 minutes before slicing.
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.
Spoon beet mixture onto plates and arrange sliced pork on top. Spoon a generous amount of walnut sauce over.