For the Brown Butter: Place the butter in a saucepan over high heat; melt and cook until the milk solids caramelize (they will caramelize and will smell like toasted hazelnuts; this is also known as beurre noisette).
Cool the butter to room temperature.
For the Ice Cream Base: Make the ice cream base according to the Classic Ice Cream Method. Add the brown butter to the ice cream base when it is made and it is still hot. Add the brown butter in liquid form, not when it is solid. Whisk thoroughly to completely incorporate the brown butter.
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.