Grease a 4- to 6-quart slow cooker with teaspoon butter. Add potatoes, apple juice, remaining butter, brown sugar, salt, cinnamon, and ginger.
Cover; cook on low 6 hours until largest pieces crush easily against crock. Keep lid closed most of cooking time.
Mash about three-quarters in cooker, leaving soft chunks. Fold in warm milk; smooth surface.
Toast walnuts in a dry skillet over medium 3 to 4 minutes, stirring, until fragrant. Scatter over potatoes.
Cover; cook on low 10 minutes more. Serve hot from cooker.