Place the sweet potatoes and 1 tablespoon of the salt in a large pot. Add enough cold water to cover. Bring to a boil. Reduce heat and simmer until the potatoes are tender, about 20 minutes.
Meanwhile, in a large saucepan, over medium-low heat, melt 2 tablespoons of the butter. Add the onion and cook for 10 minutes. Add the parsnips, celery, fennel, Brussels sprouts, parsley, pepper, and the remaining salt and toss. Add the broth, increase heat, and simmer until the vegetables are tender, 15 to 20 minutes. Add the spinach or Swiss chard (if using) and stir until wilted. Remove from heat. Using a slotted spoon, transfer the vegetables to a casserole or 9-inch pie plate.
Drain the potatoes and return to pot. Add the remaining butter and mash until smooth. Spread the mashed potatoes over the vegetables.
Set broiler on high. Broil until lightly browned, 3 to 5 minutes. Sprinkle with the nutmeg.