Whisk together milk, sugar, egg yolks, vanilla, cream, and cake mix in a saucepan until well blended. Cook over medium-low heat until mixture reaches 160 degrees F (70 degrees C), stirring frequently. Remove from heat and place in the refrigerator or freezer until liquid is cold.
Pour the chilled mixture into an ice cream maker and freeze according to manufacturer's directions until it reaches soft-serve consistency. Transfer ice cream to a one- or two-quart lidded plastic container; cover surface with plastic wrap and seal. For best results, ice cream should ripen in the freezer for at least 2 hours or overnight.