Place four 8-ounce ramekins in a 3-inch-deep baking pan large enough to allow at least an inch of space on all sides. Bring a large kettle or pot of water to a boil, to make a bain marie for the ramekins. Preheat the oven to 375°F.

In a 1-quart pot, put the coconut milk, lemongrass, lime leaves, and ginger. Bring to a boil, cover, and reduce the heat to a low simmer. Cook for 20 minutes. Pour the hot coconut-milk mixture through a strainer placed over a large bowl. Discard the lemongrass, leaves, and ginger. Mix the palm sugar into the hot coconut milk to dissolve it. Stir in the cooked rice and dried fruit. In a small bowl, whisk the eggs and vanilla. When the coconut milk has cooled to body temperature, whisk in the egg mixture.

Portion the mixture among the ramekins, making sure to distribute the solids and liquids evenly. Put them in the baking pan and pour the hot water in around them, so that it comes halfway up the sides. Carefully put the pan in the oven and bake for 45 minutes. Check for doneness by jiggling one ramekin; if the sides are puffed and the center barely moves, they are done. Take the ramekins out of the water to cool on a rack. Serve warm or chilled.