Season the pork with 1 teaspoon kosher salt and the black pepper. In a small bowl, whisk together the grated apple or pear, soy sauce, miso, brown sugar, rice vinegar, sesame oil, and chicken broth until mostly smooth.

Heat a wide, heavy skillet or Dutch oven over medium-high heat for 2 minutes, then add the neutral oil. Sear the pork in two batches, turning occasionally, until browned on several sides, 6 to 8 minutes per batch. Transfer the browned pork to a plate.

Reduce the stovetop heat to medium. Add the onion to the same pan with the remaining 1/4 teaspoon salt and cook, scraping up browned bits, until softened and lightly golden, 5 to 6 minutes.

Stir in the garlic and ginger and cook for 30 seconds. Pour in the soy-miso apple mixture and bring it to a strong simmer, scraping the bottom of the pan to release any remaining fond.

Return the pork and any accumulated juices to the pan. Adjust the heat to low, cover, and simmer gently on the stovetop for 28 to 32 minutes, turning the pork once or twice, until the pieces are tender but not falling apart.

Uncover the pan and stir in the mushrooms and cabbage. Increase the heat to medium-low, cover again, and cook until the vegetables soften and the pork is very tender, 8 to 10 minutes.

Use tongs to transfer the pork and vegetables to a warm shallow bowl, leaving as much sauce in the pan as possible. Bring the sauce to a lively simmer over medium-high heat. Stir the cornstarch and cold water together, then whisk the slurry into the sauce and simmer until glossy and lightly thickened, 2 to 3 minutes.

Return the pork and vegetables to the skillet and turn gently to coat in the reduced sauce. Keep over low heat for 2 minutes so the glaze clings to the pork.

In a small bowl, toss the scallions, matchstick pear or apple, and sesame seeds. Spoon the braised pork and vegetables over cooked rice, udon noodles, or your chosen starch, then finish with the crisp scallion-pear garnish.