Cook sweet potatoes until soft, either in oven or boiling them (see notes below).
Place peeled sweet potatoes in a mixing bowl and add brown sugar, salt, vanilla, eggs, and melted butter. Mix until well blended. Transfer filling to a greased 2 qt baking dish.
Bake in a 350 degree F oven for 30 minutes. Remove from oven and sprinkle mini marshmallows on top. Return to oven for 10 minutes, or until lightly browned.
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Notes
To bake sweet potatoes: Place unpeeled sweet potatoes on baking sheet and bake in a 400 degree F oven for about an hour (place a piece of aluminum foil on the baking sheet for easy clean up)! Once done, slice in half and scoop out center.
You could also boil the sweet potatoes. While this is a faster option, it's more hands on. Don't forget, for the boil option you'll need to peel the potatoes first!