In a large saucepan make the rice pudding. First put the rice in the pan. Whisk together milk, sugar, egg, salt, and vanilla. Pour over rice.
Cover and bring to a boil over high heat. Once boiling, reduce heat to low and simmer 20 to 30 minutes until rice is cooked and absorbed most of the liquid, leaving a creamy sauce.
Meanwhile, make the salted caramel sauce. In a medium-size saucepan over medium heat, melt the sugar. Stir constantly with a rubber spatula, scrapping the sides as you go. Once the sugar melts and turns a golden brown, add in butter until melted. Pour in heavy cream and bring to a simmer, without increasing the heat. Simmer 90 seconds. Remove from heat and stir in salt.
Stir 2/3rd of the salted caramel mixture into the rice pudding, reserving the remaining 1/3rd for drizzling over the top.