Combine the milk and cream in a saucepan over medium heat. Heat until bubbles form around the edges. In a medium bowl, whisk together the egg yolks and sugar until smooth and well combined. Slowly add the warm milk mixture to the egg yolk mixture, whisking constantly to prevent curdling.
Pour the mixture back into the saucepan. Cook over medium heat, stirring constantly, about 5-8 minutes, until the mixture is thickened and coats the back of a spoon (about 175 degrees F on an instant read thermometer). Pour the liquid through a fine mesh sieve into a bowl. Stir in the vanilla extract. Cover with plastic wrap and chill in the refrigerator until completely chilled.
Once the mixture is well chilled, pour into an ice cream maker and freeze according to the manufacturer’s instructions. Once the mixture is softly frozen, transfer half of it to a storage container. Add half of the chopped Oreo pieces and fold in gently with a rubber spatula. Add the remaining ice cream and Oreo pieces to the container, and fold once more until the mixture is evenly combined. Freeze until completely hardened.