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1
Puree the cucumber in a blender or food processor. You should have a scant 2/3 cup of puree, some of which will be 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.)
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2
Put the mint leaves and the sugar into the food processor bowl. Process until the mint is finely ground. (You could also grind them together with a mortar and pestle. )
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3
In a medium saucepan, warm the milk and heavy cream over medium heat. Stir occasionally.
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4
Whisk the egg yolks until smooth, then add the sugar and mint mixture and whisk until thoroughly combined. When the milk mixture steams and is hot but not simmering, whisk 1/2 cup of the hot milk into the egg mixture to temper the egg yolks.
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5
Take the saucepan off the heat and stir in the tempered egg mixture.
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6
Return the saucepan to the stove over medium heat. Add the honey and salt. Cook, stirring, until the mixture is between 175 and 180 degrees, or thick enough to coat the back of the spoon. This will take 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. )
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7
Strain the cucumber through a fine mesh strainer into the custard. Stir in the vanilla extract.
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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.
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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 beginning look like ice cream), about 5 minutes before it's ready, pour in the white chocolate.
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10
It's good, right?!
