The first time I enjoyed apple mashed potatoes was at a trendy New York restaurant called Barca 18. They were delicate, replete with sauteed apples. I just knew I had to develop my own version of the dish. I went all out and caramelized the apples before folding them into the potatoes. It's easy to do, and the sugar adds its mellow sweetness when it's combined with the rest of the ingredients.
Place the potatoes in a large pot and add cold water to cover. Salt the water and bring to a boil over high heat. Reduce the heat to medium-low and simmer, partially covered, until the potatoes are very tender, about 20 minutes.
While the potatoes are cooking, prepare the apples: Melt the 3 tablespoons of butter in a large nonstick skillet over low heat. Add the apples and cook, stirring, until they have softened, about 10 minutes. Sprinkle the sugar over the apples and continue to cook, shaking the pan, until they are caramelized but not burned, 2 to 3 minutes.
Set the apples aside.
Heat the half-and-half in a small saucepan over medium-low heat until bubbles just begin to form around the edges (do not let it boil). Set the pan aside.
Drain the potatoes well and return them to the pot. Place the pot over very low heat and shake it for about 1 minute to dry the potatoes well.
Transfer the potatoes to a bowl, and add the 8 tablespoons of butter. Using a ricer, a potato masher, or a large fork, mash the potatoes with the butter. Stir in the half-and-half, and season generously with coarse salt and black pepper. Fold in the reserved apples. Serve immediately.