I am indebted to my friend David Lebovitz for the method in this ice cream. He is a master ice cream maker, and his trick for leaving aside a cup of cream to quickly cool the hot custard really works well.
Put the milk, salt, sugar and 1 cup of the heavy cream in a pot and heat over medium-low heat, stirring often, until everything combines. Add the mint leaves and stir to get them all wilted in the hot milk. Turn off the heat, cover the pot and let the milk return to room temperature. When it hits room temperature, chill overnight in the fridge.
The next day, strain the mixture through a fine-meshed sieve. It's OK if a few tiny mint leaf specks get in there. Reheat the mixture over medium-low heat. Whisk the egg yolks in a bowl. In another bowl, pour in the other cup of heavy cream and set the fine-meshed strainer over it.
When the mint-cream mixture is hot, about 150°F, you are ready to temper your eggs. With one hand whisking the eggs, slowly pour in a ladle of the hot cream. Do this three times total, whisking and pouring the whole way. Then pour the egg-cream mixture into the pot and stir to combine.
Cook the mixture slowly, stirring often, until it thickens enough to coat the back of a spoon, about 160°F. Pour it through the strainer into the bowl with the other cup of heavy cream; doing it this way cools the mixture quickly. Chill everything down and churn according to your ice cream maker's directions.
Notes
Note that prep time does not include the time it takes to churn or chill the ice cream.