In a large bowl, whisk the egg yolks and sugar until frothy.
In a medium sauce pot, stir the milk, mascarpone and vanilla over medium heat. As the ingredients heat up, use a whisk to help incorporate the mascarpone cheese into other ingredients until smooth.
Once the liquid is just starting to simmer, turn the heat to low. Ladle half of the milk mixture into the yolks while whisking. Pour the remaining mixture into the bowl while whisking, then return the ingredients to the saucepan.
Stir constantly with a spatula for several minutes until the ice cream base thickens slightly and coats the back of the spatula. Transfer to a clean container and cover with plastic wrap, pressuring directly against the mixture to prevent a skin from forming. Chill for several hours (or overnight), until very cold. Note: you can speed up this process by placing the saucepan in an ice bath and stirring for about 15 minutes to cool it down before transferring to the clean container.
Re-whisk the ice cream base (the cheese may have separated), then prepare in an ice cream maker according to manufacturer instructions. Transfer into a freezer-safe container, layering it with the chopped sandwich cookies. Place in the freezer until the ice cream has firmed up and is ready to serve.
You can use any type of dairy milk with this recipe. Mascarpone cheese will add plenty of fat; heavy cream is not necessary and might actually be too rich.