In a medium saucepan, mix rice with milk and salt. Bring to a boil over medium-high heat, then lower heat and simmer, uncovered, 10 minutes.
Meanwhile, preheat broiler to high. In a small bowl, whisk together vanilla, egg yolks, and 1/3 cup sugar. Slowly stir egg mixture into rice and cook until super creamy and thicker, about 1 minute.
Divide rice pudding among four (4-oz.) ramekins, then top each with 1 to 2 tablespoons sugar (enough to cover thickly). Set on a baking sheet and slide under broiler. Broil puddings, watching closely, until tops are caramelized and golden, 4 to 5 minutes. (Alternatively, caramelize sugar using a kitchen-style blowtorch.)