Melt the butter with the cinnamon stick in a heavy medium skillet over medium heat until melted, about 2 minutes. Remove from the heat and set aside.
Cook the gnocchi in a large pot of boiling salted water until all the gnocchi rise to the surface, about 3 minutes. Discard the cinnamon stick from the butter sauce and reheat the sauce over low heat. Using a slotted spoon, transfer the cooked gnocchi to the hot cinnamon butter in the skillet. Sprinkle with the sugar and cinnamon. Toss to coat and add more sugar, to taste.
Spoon the gnocchi and butter sauce into shallow bowls and serve immediately.