Put the potatoes in a large saucepan and cover with cold water by 1 inch. Season generously with salt and bring to a boil. Reduce the heat to medium and simmer until very tender, about 15 minutes.
Meanwhile, cook the bacon in a large skillet over medium heat until crisp, 7 to 8 minutes; transfer to a paper towel-lined plate.
Drain the potatoes and return to the saucepan over low heat; cook until any remaining water evaporates, about 2 minutes. Add the butter, then gradually mash in the sour cream with a potato masher. Stir in the bacon and chives, reserving 1 tablespoon of each. Transfer to a serving dish; top with cheddar and the remaining bacon and chives.