In a Dutch oven or large stock pot, heat the olive oil over medium-high heat until shimmering. Sprinkle pork generously with salt on all sides. Add to pot and sear without moving until a dark brown crust forms on one side, about 5 to 6 minutes.
Flip pork over and continue browning on the second side until a dark crust forms, about 4 to 5 minutes longer. Remove from pot and transfer to slow cooker. Pour applesauce over the top.
Cover and cook on HIGH for 4 to 6 hours or LOW 8 to 10 hours. Using two forks, shred the pork. Pour barbecue sauce on top if using.
Cover and cook until barbecue sauce is heated through, about 30 minutes longer. Season to taste with salt and pepper.