Whisk together the eggs and sugar until the yolks are a pale yellow. Combine egg mixture and remaining ingredients in a saucepan.
Cook over medium-low heat, stirring frequently, making sure not to boil. Remove from heat when the mixture is thick enough to coat the back of a spoon.
Strain through a fine mesh strainer into a bowl, cool to room temperature. Chill in refrigerator overnight.
Churn in an ice cream maker.