Put the mint leaves and the sugar into the food processor bowl. Process until the mint is finely ground. If you do not have a food processor, put the sugar and mint leaves into a bowl and use a pestle to grind the leaves into the sugar.
In the medium saucepan, warm the milk, heavy cream, and vanilla bean (not the vanilla extract) over medium heat. Stir occasionally.
In the medium bowl, whisk the egg yolks until smooth, then add the sugar and mint mixture and whisk until combined.
When the milk mixture steams and is hot but not simmering, whisk 1/4 cup of the hot milk into the egg mixture, then whisk in another 1/4 cup of the hot milk. This helps temper the eggs so they don't scramble when you add them to the hot milk. Take the saucepan off the heat and stir in the tempered egg mixture.
Return the saucepan to the cooktop over medium heat. Add the honey and salt. Cook, stirring, until the mixture is thick enough to coat the back of the spoon, about 5 minutes. Do not allow the mixture to boil. When you draw your finger across the spoon, it should leave a clear mark through the custard. If you choose to use vanilla extract instead of the vanilla bean, add the extract now.
Quickly set the saucepan into the large bowl filled with ice water to cool the custard. You can also pour the custard into another heat proof pan if you prefer. Stir the custard for a few minutes. Cover and refrigerate until completely cold, at least 1 hour and up to 24 hours.
Pour the custard through a fine mesh strainer to separate the mint leaves and the vanilla bean. Churn the custard in your ice cream freezer according to the manufacturers directions.