Place the half-and-half, basil, sugar, salt and egg yolks in a blender. Blend until completely smooth.
Transfer the mixture to a double boiler, and heat, stirring constantly, until the eggs just begin to thicken and the sugar has dissolved. Remove from the heat and strain through a fine-mesh sieve into a metal bowl. Set the bowl in an ice water bath to rapidly chill.
Churn in an ice cream machine according to manufacturer's instructions. Freeze until ready to use.