Brown the butter. In a medium pan, melt the butter on medium heat. Increase to medium-high heat. Stirring constantly, simmer the butter until browned and dark brown flecks remain. It should smell fragrant, like hazelnuts. Set aside.
In a medium saucepan, heat milk and cream on medium-high heat until just boiling, stirring occasionally. Remove from heat.
While milk cooks, whisk together egg yolks, sugar and salt. Slowly, whisk in half of the milk mixture until smooth. Return saucepan to heat, then whisk in the egg mixture. Stir until the mixture thickens and coats the back of a spoon.
Prepare an ice bath. Add ice to a large mixing bowl, then set a medium mixing bowl on top of the ice. Through a fine mesh sieve, pour ice cream mix the smaller bowl in the ice bath. Stir in the browned butter. (Pour the butter through a sieve if you prefer a smoother ice cream.) Refrigerate for 1 hour.
Prepare ice cream according to your ice cream maker's instructions. Near the end, add moon pie cookie pieces. Freeze in a freezer-safe container several hours or overnight before serving.