Heat the milk, cream, sugar, cake mix, and salt in a saucepan over medium heat, whisking until the sugar is dissolved completely. You want the mixture to be warm, but not simmering.
Place the egg yolks in a large bowl and whisk. Add about 1/2 cup of the warmed milk mixture to the yolks, stirring constantly, to temper the eggs. Then add the egg mixture into the warmed milk mixture in the saucepan. Heat over medium heat and, stirring often, cook until the mixture has thickened and coats the back of a spoon.
Remove from the heat and pour mixture into a large bowl. Stir in the vanilla extract. Cover the bowl and refrigerate overnight or for at least six hours.
After refrigerating, add the mixture to the ice cream maker and churn for 25 minutes. When you have a few minutes left, add cake cubes and sprinkles.
Place the ice cream in a container and freeze for 4 to 6 hours.