Slip the potatoes into a large pot of cold salted water and bring to a boil. Reduce the heat and simmer until tender, 15 to 20 minutes.
Meanwhile, heat the butter and half-and-half in a small pot until the butter melts and the mixture is hot.
Once the potatoes are cooked, drain well in a colander and then return them to the pot. Turn the heat back on to low and stir the potatoes to "dry" them. Mash the potatoes until smooth and stir in the hot half-and-half and butter. Add the shredded cheese by the handful, stirring to melt, and season generously with salt and pepper. Stir in the chives. Serve immediately.