In a medium mixing bowl, beat the sugar into the eggs until thickened and pale yellow (a couple of minutes). Set aside.
Bring the milk to a simmer in a heavy medium saucepan. Slowly beat the hot milk into the eggs and sugar. Pour the entire mixture back into the pan and place over low heat. Stir constantly with a whisk or wooden spoon until the custard thickens slightly. Be careful not to let the mixture boil or the eggs will scramble. Remove from the heat and beat in the peanut butter. Pour the hot custard through a strainer into a large, clean bowl. Allow the peanut butter custard to cool slightly. Stir in the cream and vanilla. Cover and refrigerate until cold or overnight.
Stir the chilled custard then freeze in your ice cream machine according to the manufacturers instructions. When finished, the ice cream will be soft but ready to eat. For firmer ice cream, transfer to a freezer-safe container and freeze for at least 2 hours.
Double Peanut Butter Ice Cream: Add 1 cup of crumbled peanut butter cups to the machine when the ice cream is semifrozen. Allow the machine to mix in the candy.