Combine 1/4 of the onion, 1/2 of the apple, and the garlic in the bowl of a food processor. Process until smooth. Transfer to slow cooker along with the apple cider and water.
Season the roast with salt and pepper. Combine the ginger and flour in a shallow container. Gently press the roast onto the flour mixture to coat all sides. Brush off any excess flour. Heat the oil in a large, heavy skillet. Brown the pork in the hot oil, turning until golden on all sides. Place the browned roast in the slow cooker and scatter the celery and the remaining onion and apple on top. Cook on low for 4 hours, then add carrots. Cook for an additional 3 hours.
Remove the cooked roast and carrots to a heated platter. If desired, thicken the cooking juices from the slow cooker to make a gravy.