Place the whole potatoes in a saucepan with water to cover. Bring to a boil and cook at a low boil until they are soft, about 45 minutes. While still warm, peel the potatoes and pass them through a vegetable mill onto a clean pasta board.
Make a well in the center of the potatoes and sprinkle all over with the flour, sugar, and cinnamon. Bring the ingredients together to form a homogenous dough. Knead for 4 minutes, until the ball is dry to the touch. Divide the dough into 6 large balls. Roll each ball into 3/4 inch-diameter ropes and cut the ropes into 1-inch-long pieces. Flick the pieces off of a fork or along the concave side of a cheese grater, to score the sides. Drop the dough pieces into the boiling water and cook until they float to the surface, about 1 minute. Drain the gnocchi and transfer to a mixing bowl. Toss with the melted butter, grate Parmigiano-Reggiano over, and serve immediately.