Beat eggs until light and foamy. Add sugar amd salt. beating until thick.
Combine egg mixture and 2 cups light cream in a large saucepan. Stirring constantly, cook over low heat until mixture coats the back of a metal spoon and reaches 160° on a candy thermometer. Remove from heat.
Stir in remaining light cream; add rum or extract and water. Chill several hours. Sprinkle with nutmeg before serving. Garnish with cinnamon sticks.