Arrange sweet potato rounds in a wide heavy skillet with a lid, overlapping slightly if necessary. Add butter, brown sugar, water, cinnamon, nutmeg and salt.
Bring to a gentle simmer over medium heat, then cover and cook for 15 minutes. Shake the pan occasionally rather than stirring hard.
Turn the rounds carefully with a thin spatula, cover again and simmer for 8 to 10 minutes until a knife enters easily.
Uncover and simmer for 5 to 8 minutes, spooning syrup over the rounds, until the liquid thickens and coats the potatoes.
Remove from the heat and gently fold in vanilla. Let stand for 5 minutes so the syrup thickens slightly before serving.