Place the potatoes in a large pot and cover with cold water. Bring to a boil and add 2 1/2 tablespoons salt. Reduce heat and simmer until just tender, about 15 minutes.
While the potatoes are simmering, peel and thinly slice 5 medium shallots. Heat 1 tablespoon olive oil in a medium skillet over medium heat. Add the shallots and cook, stirring occasionally, until soft and golden brown, about 15 minutes.
Drain the potatoes and return to the pot. Add the butter, buttermilk, cream, nutmeg, 1/2 teaspoon salt, and 3/4 teaspoon pepper. Mash the potatoes with a handheld masher. Fold the shallots into the mashed potatoes before transferring to a serving dish.
Make-Ahead Note: The potatoes can be made up to 2 hours in advance and kept warm in a double boiler or Crock-Pot. If the potatoes are chilled and then reheated, they will turn dry and pasty.