Preheat the oven to 325°F (163°C).
In a large bowl, cream together 1 cup unsalted butter and 2/3 cup granulated sugar until light and fluffy.
Add 1/4 teaspoon salt and gradually mix in 2 cups all-purpose flour until a dough forms.
Press the dough evenly into a 9×13 inch baking pan lined with parchment paper.
Bake the shortbread for 30 to 35 minutes, or until the edges are lightly golden.
Remove from oven and let cool completely, then cut into small bite-sized pieces and set aside.
In a medium saucepan, combine 2 cups heavy cream, 1 cup whole milk, and 3/4 cup granulated sugar. Heat over medium heat until the sugar dissolves and the mixture is hot but not boiling.
In a separate bowl, whisk 5 large egg yolks until smooth.
Slowly pour about 1 cup of the hot cream mixture into the egg yolks while whisking constantly to temper the eggs.
Pour the tempered egg yolk mixture back into the saucepan with the remaining cream mixture.
Cook over low heat, stirring constantly with a wooden spoon, until the mixture thickens enough to coat the back of the spoon (about 5-7 minutes). Do not boil.
Remove from heat and stir in 1 tablespoon pure vanilla extract.
Strain the custard through a fine mesh sieve into a clean bowl.
Allow the custard to cool to room temperature, then cover and refrigerate for at least 4 hours or overnight.
Once chilled, pour the custard into an ice cream maker and churn according to the manufacturer’s instructions until it reaches a soft-serve consistency.
Fold in the shortbread cookie pieces gently.
Transfer the ice cream to an airtight container and freeze for at least 1 hour to firm up before serving.