Place egg yolks in the bowl of an electric mixer with the whip attachment. Whip on high until light and fluffy. Set aside.
Prepare an ice bath to cool the custard by placing a smaller bowl inside a larger bowl that has been filled with ice and water. Set aside.
In a medium saucepan over medium heat, combine the heavy cream and the milk with half of the basil. Bring to a simmer.
Remove from the heat and allow to steep for 10 minutes. Strain basil from the cream mixture. Discard leaves. Return cream mixture to the saucepan.
To the cream mixture, add the sugar. Return pan to the heat. Stir to dissolve the sugar.
Once the mixture begins to come to a simmer (look for steam rising from the surface of the cream mixture), use a ladle to gradually add some of the hot cream mixture to the beaten egg yolks. Whisk to incorporate. Transfer the egg yolk mixture to the saucepan.
Continue to cook over medium to medium-low heat, stirring constantly with a wooden spoon. Check the consistency of the custard by running your finger down the back of the wooden spoon through the custard. If your finger leaves a trail, the custard is the proper consistency. Remove the custard from the heat and add the remaining basil leaves. Use an immersion blender to blend. Strain the custard through a fine mesh strainer into the smaller bowl of the ice bath. Stir the custard to facilitate cooling. When the custard has cooled, transfer it to a storage container and refrigerate overnight.
When ready to freeze the ice cream, spin the deeply chilled custard in an ice cream freezer according to the manufacturer's instructions.