Add the milk, cream, chopped doughnut and beer to a pan over medium-high heat. Cook, stirring frequently until the milk starts to bubble around the edges and the doughnut has broken down. Remove from heat, allow to cool slightly.
Add the yolks to a mixing bowl with the granulated sugar, powdered sugar, vanilla, and salt. Stir until well combined.
Slowly add the milk mixture while whisking continuously.
Add to an airtight container. Refrigerate until chilled, about 3 hours.
Add to an ice cream maker, churn according to manufactures specifications until a soft-serve consistency (in a KitchenAid ice cream maker, this takes about 15 minutes).
Transfer to a freezer container, freeze until set, about 2 hours.