Preheat oven to 350°F (175°C) and grease a 2-quart casserole dish.

Bring milk to a boil in a large saucepan. Stir in rice and salt, reduce heat to medium-low, and cook while stirring frequently until the rice is tender, about 18 minutes.

In a bowl, beat egg yolks, ½ cup sugar, butter, and vanilla extract until combined. Stir in 1 cup of the hot rice mixture until well mixed.

Pour the egg mixture into the saucepan with the remaining rice mixture. Stir constantly until the mixture thickens and comes to a boil, about 1 minute. Pour into the prepared casserole dish.

In a separate bowl, beat egg whites until soft peaks form. Gradually add 3 tablespoons sugar and continue beating until stiff peaks form. Spread the egg white mixture over the rice mixture, ensuring it touches the edges of the casserole dish.