Put the sweet potatoes in a large pot and cover with cold water by 1 inch. Bring to a boil, then simmer for 15 to 18 minutes until easily pierced; drain well and let steam dry for 3 minutes.
Heat the oven to 350°F and butter a 2-quart baking dish with the measured teaspoon butter.
Mash the hot potatoes with the remaining butter, evaporated milk, brown sugar, vanilla, salt and cinnamon until mostly smooth. Spread evenly in the dish.
Bake for 25 minutes until the center is hot. Scatter the marshmallows in a single loose layer over the surface.
Bake for 5 to 7 minutes more, watching closely, until the marshmallows puff and turn golden. Let stand for 5 minutes before serving.