Place the potatoes and 1 tablespoon of salt in a 4-quart saucepan with cold water to cover. Bring to a boil, lower the heat, and simmer covered for 25 to 35 minutes, until the potatoes are completely tender. Drain.

In a small saucepan, heat the half-and-half and butter. Transfer the potatoes to an electric mixer fitted with the paddle attachment and mix them for a few seconds on low speed to break them up. Slowly add the hot cream and butter to the potatoes, mixing on the lowest speed (the last quarter of cream and butter should be folded in by hand). Fold in the sour cream, Parmesan, the remaining salt, and pepper; season, to taste, and serve immediately. If the potatoes are too thick, add more hot cream and butter.

Keep the smashed potatoes hot in a heat-proof bowl set over simmering water.