While the cream is warming up, whisk egg yolks and sugar together in a bowl. Make sure they're well combined.
When the cream is just starting to simmer (don't boil it), slowly temper it into the yolks while whisking.
Pour the mixture back into the saucepan and turn the heat down to medium low. Stir in a figure 8 motion with a wood spoon for several minutes until the mixture coats the back of the spoon. Remove from the heat. Continue to stir over an ice bath until cool.
Puree the basil and cream in a blender. Allow the mixture to infuse for at least an hour in the fridge, then strain the basil cream through a fine mesh strainer into your ice cream machine.
If using, add a few drops of green food coloring.
Whisk in the olive oil, 1/4 - 1/2 cup.
Make ice cream according to the manufacturer's instructions.