Combine first 4 ingredients in a Dutch oven. Gradually stir in scalded milk. Cook 20 minutes over medium heat, stirring constantly with a wire whisk.
Combine eggs and 1 cup sugar. Gradually stir a small amount of hot milk mixture into egg mixture. Add egg-milk mixture to remaining hot milk mixture; stir constantly. Remove from heat; cool completely. Chill at least 2 hours.
Add whipping cream and vanilla to chilled mixture, stirring with a wire whisk to combine.
Pour mixture into freezer can of a 1-gallon hand-turned or electric freezer. Freeze according to manufacturer's instructions. Let ripen 1 1/2 to 2 hours.
Chocolate Ice Cream: Combine four (1-ounce) squares of semi-sweet chocolate and 1/4 cup hot water in top of a double boiler; place over boiling water, stirring until chocolate melts. Proceed as directed for Vanilla Ice Cream; add melted chocolate mixture to scalded milk-sugar mixture after the latter has cooked 20 minutes.