In a large saucepan, whisk together sugar, fresh lemon juice, egg yolks, and lemon zest over medium heat. Cook for 1 minute.
Add heavy cream and half n half. Stir constantly with a whisk and cook for 5-6 minutes over medium-low heat. Whisk constantly. Once it starts to bubble, remove from heat.
Place in a bowl and lightly cover with plastic wrap and chill in the refrigerator.
Once chilled, place in ice cream maker and churn until ice cream is creamy.
Lemon Ice Cream Cookie Sandwiches:
Preheat oven to 350 degrees.
In a large bowl, cream butter and sugar for 4 minutes until light and fluffy. Add eggs, vanilla, lemon juice, and lemon zest and stir to combine.
Stir in flour, baking powder, baking soda, and salt and mix just until combined.
Drop dough onto light colored baking sheet.
Bake for 9-11 minutes. Sprinkle with sugar or white sparkling sugar.
Once cooled, scoop lemon ice cream on the bottom of lemon sugar cookie and top with another cookie.