In a saucepan, combine the milk, vanilla, sugar, cinnamon, and nutmeg and heat over medium heat until the sugar is dissolved. Turn the heat to low to keep the mixture warm.
In a large saucepan, melt the butter. Add the rice and stir to coat. Add the Champagne and cook, stirring until the liquid is absorbed. Ladle the milk into the rice 1/2 cup at a time, stirring with each new addition, about 35 minutes.
Remove from the stove and stir in the remaining milk, if desired, to achieve a creamier consistency. Divide among dessert bowls and sprinkle with the Demerara sugar.