Mix milk, rice, 1/2 of the sugar, vanilla extract, and salt together in a saucepan over medium-high heat. Bring to a boil.
Blend remaining sugar with whole eggs and yolks in a bowl to make a creamy custard.
Reduce heat of the milk mixture. Let simmer, uncovered, until thickened, 30 to 40 minutes. Fold in the custard. Heat until thickened again, about 5 minutes more.
Beat cream in a chilled glass or metal bowl with an electric mixer until soft peaks form.
Remove the milk and custard mixture from heat. Let cool, about 10 minutes. Fold in the whipped cream. Pour pudding into a serving dish, cover, and refrigerate until chilled, at least 2 hours.