Combine 8 cups whole milk, sugars, and rice in a large saucepan over medium-low heat. Simmer, covered, for 1 hour, stirring frequently. The rice should be tender. Remove the pan from heat and let sit for 10 minutes.

Meanwhile, cook apples in a small frying pan with butter under tender, about 10 minutes. Set aside.

Whisk together the eggs, remaining ¼ cup milk, vanilla, and salt in a medium bowl.

Stir into the rice mixture and return the pan to low heat, stirring constantly, for 2 minutes.

Mix in the cooked apples and cinnamon.

Pour the pudding into a 9x13-inch dish and cover with plastic wrap, folding back the corners to allow steam to escape. Cool to room temperature. Replace plastic wrap if it is wet and refrigerate until chilled.