Fill a large mixing bowl with cold water. With a swivel-bladed vegetable peeler, peel each potato and remove any "eyes," then cut it in quarters with a sharp knife and put in the bowl of water.
Put the potatoes in a large pot and add cold water to cover them. Salt the water. Over high heat, bring the water to a boil, skimming off foam that rises to the surface; then reduce the heat to maintain a simmer and cook the potatoes just until tender enough to be pierced easily with a skewer or a sharp knife tip.
While the potatoes are boiling, prepare the onions: In a deep-fryer or a deep, heavy saucepan, heat about 3 inches of oil to 375 degrees F. Lightly dust the onion rings with flour and carefully drop them into the hot oil, cooking them in batches if necessary to avoid overcrowding. Fry them until golden, 3 to 5 minutes, removing them with a slotted spoon or wire skimmer to drain on paper towels. Season lightly with salt and keep warm.
When the potatoes are done, drain them immediately. Using a ricer or a food mill, puree the potatoes into a mixing bowl. Add the nutmeg, 8 tablespoons of the butter, and the hot cream. With a wooden spoon, stir them into the potatoes. Season, to taste, with salt and pepper.
Mound the potatoes in a heated serving bowl. Put the remaining butter in a saucepan or skillet and, over medium-high heat, melt it and continue cooking just until it turns golden brown. Lightly season the butter with salt and immediately drizzle it over the potatoes.
Pile the crispy onions on top of the mashed potatoes. Garnish with parsley and serve immediately.