Preheat the oven to 375 degrees F (190 degrees C).

Mix sugar, flour, and salt together in a 3-quart pot. Slowly add milk and half-and-half and bring to a boil. Cook until slightly thickened, 2 to 3 minutes. Remove from heat. Whisk in lemon juice, lemon zest, and butter. Return to medium heat and cook, stirring constantly, for 2 minutes. Remove from heat and set custard aside.

Bake empty crust in the preheated oven for 3 minutes. Remove from oven and cool completely, about 30 minutes.

Fill cooled crust with lemon custard filling and allow pie to cool completely. Place in the freezer and freeze for 8 hours to overnight.