Place the potatoes, chicken broth, and salt in a large slow cooker. Cover and cook on high for 4-4.5 hours, or until the potatoes are fork tender. Carefully drain off any excess broth, then return the potatoes to the crock pot.
Keep the crock pot on the warm setting. Give the potatoes a quick mash with a potato masher, then drizzle in the milk, continuing to mash. Add the butter and sour cream, stirring it into the potatoes, and allowing them to soak up all the liquid, mashing a bit more, if needed.
Taste and season with additional salt, if necessary. Stir in chives, then top with a few extra pats of butter. Cover the slow cooker, allowing the butter to melt on top before serving. The potatoes can be kept on the warm setting for up to 2 hours.