Peel and dice potatoes, making sure all are relatively the same size. Place in a 6 to 7 quart heavy pot or dutch oven, add the salt, and cover with water. Bring to a boil over medium-high heat and then reduce heat to maintain a rolling boil. Cook until potatoes fall apart when poked with a fork, 15 to 20 minutes.
While the potatoes cook, melt the butter in a small sauce pan over low heat. Add the garlic and gently simmer until garlic is golden and fragrant, about 15 minutes.
Heat the half and half in a medium saucepan over medium heat until simmering. Remove from heat and set aside.
Drain the potatoes in a colander or mesh strainer then return to the pot and the heat. Toss and shake gently for 1 minute to cook off any surface moisture, then remove from heat.
Pour all of the garlic butter and the parmesan, followed by a quarter of the half and half. Begin folding gently with a rubber spatula, adding the half and half bit by bit as you go until it is all incorporated. Next, blend the potatoes with a hand held mixer until mostly smooth, 30 seconds to 1 minute. Take care to not over blend as they will become gummy. Let stand for 5 minutes to allow the mixture to thicken. Serve with additional parmesan and white pepper, if desired.