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