Preheat the oven to 325°F (165°C) and move the oven rack to the lower-middle position.
Pat the 3 pounds boneless pork shoulder dry with paper towels and season with 1 teaspoon kosher salt and ½ teaspoon freshly ground black pepper.
Heat 2 tablespoons olive oil in a large dutch oven over medium-high heat.
Working in batches, add the pork to the pot and sear on all sides, turning with tongs, 2 to 3 minutes per side. Remove the seared meat to a bowl.
Reduce the heat to medium and pour off all but 2 tablespoons of fat from the pot. Add the 1 large yellow onion and saute until softened, about 5 minutes.
Stir in the 2 garlic cloves, 1 teaspoon dried sage, 1 teaspoon dried thyme, 1 teaspoon ground mustard, ½ teaspoon ground nutmeg, and ¼ teaspoon ground cayenne pepper. Cook until fragrant, stirring continuously, about 1 minute.
Add the pork and any juices in the bowl back into the dutch oven.
Add the 4 cups low-sodium chicken broth or stock and 1 cup dry hard apple cider. Bring to a boil scraping any additional brown bits from the bottom of the pan, cover, and place in the oven to braise for 2 hours.
Remove the pot from the oven and add the 3 tablespoons apple cider vinegar, ½ pound sweet potatoes (1 large), 1 pound kohlrabi (2 medium), and 2 medium firm sweet apples. Cover and place back in the oven for thirty more minutes. Remove the pot from the oven again and add the 1 cup green beans Cover and place back in the oven and cook for an additional 30 minutes until the vegetables are tender.
Taste and season with salt and pepper as needed.