Set six 6-oz. ramekins in a baking dish. Put the sugar and water into a small, heavy saucepan and set it over medium heat. Stir to dissolve the sugar. Once the sugar has dissolved and the mixture is boiling, stop stirring and turn the heat to high. When the mixture begins to color, swirl the pan until the caramel is a deep amber brown; be careful -- the caramel is extremely hot. Take the pan off the heat and immediately pour the caramel into the ramekins, swirling them right after each pour to spread the caramel evenly over the bottom.
Heat the oven to 325°F. In a medium saucepan, stir together the milk, cooked rice, ginger slices, sugar, and salt, if using. Set the pan over high heat and bring to a boil. Once boiling, take the pan off the heat, cover, and stir occasionally until the sugar is dissolved and the milk is infused with the ginger, about 10 minutes. In a medium bowl, whisk the eggs and yolks together, whisking just to combine. Slowly and carefully pour the liquid from the rice into the eggs in a very thin stream, whisking constantly (it’s okay to leave rice behind). Pour the mixture back into the saucepan and cook over medium heat, stirring constantly with a wooden spoon, until the rice is just suspended in the custard and the mixture coats the back of a spoon, 2 to 4 minutes. Immediately remove the pan from the heat. Fish out the ginger slices and discard them. Stir in the vanilla extract.
This custard is just thick enough to hold the rice in suspension. The grains are evenly dispersed and won't sink to the bottom.
Ladle the custard into the ramekins, distributing it evenly. Pour very hot water into the baking dish so it comes halfway up the sides of the ramekins. Cover the pan loosely with foil. Bake until the custards are just about set but still slightly jiggly (they shouldn’t feel too firm), 30 to 45 minutes, depending on the thickness and depth of your ramekins. Cool the ramekins in the water bath until you can safely lift them out. Cover and refrigerate overnight (so the caramel fuses with the custard). To serve, run a sharp knife around the inside rim of each ramekin and invert it onto a small serving dish.