In a bowl, stir together the cream and milk. Add the sugar and whisk until the sugar is dissolved, 3 to 4 minutes. Test for graininess by tasting a small amount of the liquid; it should feel smooth on the tongue and there should be no sugar visible on the bottom of the bowl when it is stirred or spooned out. Continue whisking, if necessary, to ensure that the texture of the finished ice cream will be smooth. Stir in the vanilla.