Preheat oven to 300F. Pat pork shoulder dry with paper towels and season all over with salt and pepper. Heat a Dutch oven (or a large oven-safe high-walled skillet) over medium-high heat, then add 1 tablespoon butter and olive oil and swirl pan. Sear pork shoulder 3-5 minutes on all sides until a deep golden brown. Transfer to a plate and set aside.
In remaining drippings over medium heat, saute shallot until lightly browned, about 3 minutes. Add garlic and saute 1 additional minute until fragrant. Pour apple cider and beef broth into pan and use a wooden spoon to scrape up any browned bits.
Add seared pork shoulder and sage to pot and bring to a simmer over medium heat. Cover Dutch oven, place in 300F oven, and let cook 2-1/2 hours.
Add potatoes, and carrots to pot, cover, and continue roasting an additional 1 hour until pork is fall-apart tender and vegetables are tender. Remove sage sprigs and discard. Optionally, add apples to pot, cover, and allow to sit 10 minutes until apples are softened.
Melt remaining 1 tablespoon butter over medium heat. Add additional sage leaves in a single layer and sear on both sides until crispy. Drain on paper towels.
Pull apart pork into bite-sized pieces and top with crispy sage for serving. Serve warm with vegetables and enjoy!