Bring milk, Arborio rice, sugar, butter, and salt to a boil in a large saucepan over medium-high heat. Reduce heat and simmer, uncovered, stirring frequently until rice is very tender, about 1 hour. Remove from heat.

Beat eggs in a bowl, and gradually stir in about 1 cup of the rice mixture with the beaten eggs, adding about 1/4 cup of the hot rice at a time. Return the egg mixture into the saucepan, and stir in the heavy cream and vanilla extract.