Place potatoes in a large pot. Add enough cold water to cover the potatoes by about 1 inch. Add plenty of salt to the water. Bring to a boil over high heat. Once the water boils, reduce the heat so that the water is simmering. Simmer uncovered until the potatoes are very tender when pierced with a knife, about 10-12 minutes. Drain the potatoes really well; return them to the pot.

Turn the stove to low heat and gently shake the pan of potatoes over the low flame for about 1 minute, just to make sure that any excess moisture evaporates. You want really dry potatoes!

Use a potato masher, potato ricer, or food mill to mash the potatoes. Add the butter and enough half-and-half or milk until the potatoes are very creamy (you might not need all of the milk, so add it gradually and stop when the potatoes reach the desired consistency). Whisk in the sour cream and chives until completely combined. Taste and season with salt and pepper. Transfer to a serving bowl and top with additional butter or chives, if desired.