Precook the potatoes by heating enough water (about 6 cups) to cover them in a large pot over high heat until hot but well below a simmer. Turn heat down to low and add the potato slices, crushed shallots and garlic, 2 teaspoons salt, and vinegar. The potatoes will cool the water down to about 160 degrees. Cook at about this temperature, well below a simmer, for about 20 minutes. Drain and run cold water over the potato slices to cool them. Let them stand in water in the pot until you?re ready to finish cooking, or drain, cover, and refrigerate.
When ready to mash, saute the minced shallots and garlic in butter over medium heat until just soft in a heavy, medium saucepan. Stir in 2 teaspoons salt and the white pepper. Add the cream and half-and-half and heat gently. Set aside.
To finish cooking, add water to the cool potatoes if necessary, bring to a boil, and simmer until fork-tender, about 5 minutes at a simmer. Drain, remove the crushed shallots and garlic, put the potatoes back in the pot, and heat for about 1 minute to dry them out. To mash, push the potatoes once through a ricer or large-holed strainer into a medium mixing bowl. Bring the shallot and cream mixture to a boil and stir into the hot mashed potatoes. Taste and add more salt or pepper as needed. Serve immediately.
Make ahead and serve hot: Make complete mashed potatoes 1 day ahead, refrigerate overnight, cover tightly with foil and heat for 40 minutes in a 325 degree oven. Serve.