In a medium saucepan, whisk together ¾ cup of the milk, all of the half-and-half, sugar, and salt. Bring the mixture to a simmer over medium heat, whisking frequently being careful to not scorch the milk.
Meanwhile, in a separate bowl, whisk together the cornstarch and reserved ½ cup milk. Whisk thoroughly so there are no lumps. Whisk the cornstarch mixture into the milk mixture and whisk constantly until it thickens and comes to a boil, about 5 minutes. Reduce heat and cook for an additional 5 minutes until very thick and a line drawn on the back of a spoon with you finger comes clean. Stir in the vanilla extract. Pour into a bowl and chill over ice.
Let mixture chill a few hours in the refrigerator or overnight. Once chilled, whisk out any remaining lumps and freeze in ice cream machine according to manufacturer's instructions.