1 large English cucumber, peeled, seeded, and roughly chopped
2/3 cup packed mint leaves
3/4 cup granulated sugar
2 cups whole milk or half & half
1 cup heavy cream
1 tablespoon vanilla
4 large egg yolks
2 tablespoons honey
1 generous pinch of kosher or sea salt
Optional -- 1/2 to 1 cup very finely chopped white chocolate
1- purée the cucumber in a blender or food processor. You should have a scant 2/3 cup of purée, some of which is liquid. You're going to strain out the solids so this is fine. Alternatively, if you have a juicer you could simply juice the cucumber.
2- put the mint leaves and sugar into a food processor bowl. Process until the mint is finely ground. (Or you could grind them together with a mortar and pestle.)
3- in a medium saucepan, warm the milk and cream over medium heat. Sit occasionally.
4- whisk the egg yolks until smooth, then add the sugar & mint mixture. Whisk until thoroughly combined.
5- take the saucepan off the heat and stir in the tempered egg mixture.
6- return the saucepan to the cooktop over medium heat. Add honey and salt. Cook, stirring, until the mixture is 175-180 degrees F, or 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.)
7- strain the cucumber through a fine mesh strainer into the custard. Stir in the vanilla.
8- cool the custard somewhat in an ice water bath and then strain it (to strain out some of the mint leaves) into a pitcher. Refrigerate until cold,
9- pour the custard into your ice cream freezer and churn according to the manufacturer's directions. When the ice cream is starting to thicken (and look like ice cream), about 5 minutes before it's ready, pour in the white chocolate.