Prepare the ice cream mixture
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.
Chill the ice cream mixture
Fill a large bowl halfway with ice cubes and enough cold water just to cover the ice cubes. Place the bowl with the cream mixture into the larger bowl and let cool for 30 to 45 minutes. Remove the bowl with the cream mixture and place a piece of plastic wrap directly onto the surface and on top of the bowl. Refrigerate for at least 3 hours or up to 24 hours.
Churn the ice cream
Prepare an ice cream maker with at least a 1-quart capacity according to the manufacturer's instructions. Remove the plastic wrap from the cream mixture and bowl. Pour the well-chilled cream mixture into the mixing container of the ice cream maker and freeze according to the manufacturer's instructions. The timing will depend on the type of machine and the temperature of the cream mixture.
Pro-tip: Even though it seems that the ice cream will be richer, do not attempt to make it with heavy cream alone. The higher percentage of butterfat in the cream tends to solidify, causing a flaky or grainy texture.