Combine mozzarella and milk in a food processor. Blend until smooth.
Combine the mozzarella mixture, heavy cream, and oregano in a small saucepan over medium heat. Bring to a simmer, stirring often.
Whisk together sugar, egg yolks, salt, and pepper in a medium mixing bowl.
Pour 1⁄3 of the hot milk mixture over the egg mixture, stirring immediately (this allows the egg temperature to rise and prevents scrambling when added back to the pan).
Add tempered eggs to the pan with the remaining milk mixture. Stir with a wooden spoon on low heat until thick. (It should coat the back of the spoon—this is called “nappe.”)
Remove from heat and place pan in an ice bath to cool.
Once cool, strain and churn in an ice cream maker, following manufacturer instructions.