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Bring verbena, milk, cream, and 1/8 teaspoon salt to a simmer in a small heavy saucepan over medium-low heat, then remove from heat and let steep, covered, 30 minutes.
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Whisk together eggs and sugar until combined, then whisk in verbena mixture. Return mixture to saucepan and cook over medium heat, stirring constantly with a wooden spoon, until custard thickens slightly and registers 175ºF on an instant-read thermometer (do not let boil).
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Immediately strain thourough a fine-mesh sieve into a bowl, pressing on and then discarding solids. Cool custard, stirring occasionally, about 30 minutes, then chill, covered, until cold, at least 2 hours.
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Freeze custard in ice cream maker, then transfer to an airtight container and put in freezer to harden, about 1 hour.