Heat oil in a large skillet over medium-high heat. Spray slow cooker with nonstick cooking spray. Place carrots, potatoes and onion in the bottom of the slow cooker.
Season pork with salt and pepper then place in the skillet. Sear until browned on all sides (about 2-3 minutes on each side), then place on top of vegetables.
In a small bowl, whisk together garlic, apple butter, dijon, soy sauce and beef broth then pour over the pork and vegetables.
Place the lid on the slow cooker and cook on low for 8-10 hours or high for 4-6 hours.
Place pork on a serving platter and cut into slices. Use a slotted spoon to remove vegetables from the slow cooker and place on the serving platter.
Pour remaining liquid in the slow cooker into a saucepan over medium-high heat. In a small bowl, whisk together cornstarch and water to make a slurry then whisk into the saucepan. Bring to a boil and stir continuously until thickened. Pour gravy over pork and vegetables and serve.