Preheat your oven to 300°F. In a 9-quart heavy Dutch oven, combine the rice, milk, salt, sugar, and 1 tablespoon of the vanilla. Stir to combine, cover, and place in the oven.
For the first hour and a half, stir carefully every 30 minutes. Then set a timer and stir every 15 minutes for another 30 minutes — you'll start to see it coming together.
Move the pot to the stovetop over your lowest burner and take the lid off. You want barely any heat here — no bubbles, no shimmer, nothing approaching a simmer. Gradually stir in the heavy cream. From here, stir every 5 minutes for about an hour and a half to two hours, until the pudding is nearly fully thickened with just a little liquid left. It'll look almost done before it actually is — keep going.
Turn off the heat. Stir in the remaining tablespoon of vanilla and let it cool on the counter, stirring now and then. Once it's no longer steaming, cover tightly and transfer to the fridge for several hours. It will thicken further as it chills, but should stay deeply, beautifully creamy. Before serving, taste it. Add more vanilla if your heart says so. Sprinkle with cinnamon, if that's how you grew up eating it.