Use a food processor to grind the basil leaves with the sugar and one cup of the heavy cream, until the leaves are ground as fine as possible. Pour half of the mixture through a strainer into a large bowl. Add the remaining 1 cup of heavy cream.

Heat the other half of the basil mixture in a saucepan along with the whole milk and salt over medium heat. In another bowl, whisk together the egg yolks. When the basil mixture is steaming, slowly pour into the egg yolks while whisking constantly. Return the mixture back into the saucepan.

Stir the mixture constantly over medium heat with a spatula, scraping the bottom as you stir, until it is thick enough to coat the back of a spoon, about five minutes. Pour the custard through the strainer into the bowl with the cream mixture. Zest the lemon directly into the custard, then cool over an ice bath while stirring.

Chill completely, then freeze in an ice cream machine according to the manufacturer's instructions. Let harden in the freezer for a few hours before serving.