Heat the half and half and sugar in a saucepan over medium-low heat. Meanwhile, whisk the egg yolks in a bowl until pale yellow and thick, about 5 minutes. Temper the egg yolks by splashing in a very small amount of the warm half and half mixture while whisking constantly. Slowly pour the tempered egg yolks into the saucepan with the half and half and cook over medium-low heat, stirring constantly, until thick enough to coat the back of a spoon, about 5 minutes.
Add the heavy cream into a large bowl along with the vanilla bean paste then pour in the egg mixture and stir to combine. Cover the bowl and let chill for at least 6 hours in the refrigerator.
In a large Ziploc bag, crush chocolate sandwich cookies with a rolling pin. You don't want to completely pulverize them, you want a variety of pieces.
Once ice cream based is chilled, pour the mixture into the bowl of the ice cream maker and freeze according to the manufacturer's instructions. Add the crushed cookies to the ice cream before it is done churning.
Transfer ice cream to a freezer-safe container and freeze overnight before serving.