Heat cream and milk together in a 4 qt. pot. When liquid boils, remove from heat, add verbena leaves, and cover. Steep about 2 hours to develop flavors.
Strain liquid and return to pot; discard leaves.
Heat liquid over high heat until it reaches a simmer.
Whisk together yolks and sugar just as liquid reaches a simmer. Working in 1/2 cup increments, add hot liquid to yolk and sugar mixture until mixture is warm and the consistency of milk.
Pour custard into pot and cook over high heat, stirring constantly, until slightly thickened, about 5 minutes. Do not let boil.
Strain custard into a bowl and nestle in another bowl filled with ice water to chill, stirring occasionally, until cold to the touch.
Churn custard in an ice cream maker according to manufacturers' instructions. Freeze in an airtight container until firm, about 4 hours.
