For the Infused Milk: Grind the espresso right before adding it to the milk. Steep the ground espresso and cardamom pods in the milk until the milk is infused, about 10 minutes. Strain the milk through a fine-mesh strainer.
For the Ice Cream Base: Make the ice cream base according to the Classic Ice Cream Method.
Strain the base through a fine-mesh strainer and cool over an ice bath. Age the base under refrigeration overnight.
Once the base has aged, churn to the desired consistency.
Harden for at least 2 hours and up to 4 hours. Reserve until needed.