Bring the coconut milk, water, and 1/4 cup of the sugar to a boil in a small saucepan. Stir in the rice. Reduce heat, cover, and simmer gently until the rice is tender and the liquid is absorbed, about 20 minutes.
Meanwhile, warm the half-and-half in a small saucepan over medium heat (do not allow it to boil). Whisk together the eggs, vanilla, and the remaining sugar in a medium bowl. Whisking constantly, slowly pour the warm half-and-half into the egg mixture. Skim off and discard any foam that rises to the surface. Stir in the rice. Transfer to an 8-by-8-inch baking dish. Place the dish in a larger roasting pan or baking dish and transfer to oven rack. Carefully add enough hot water to the larger pan to reach halfway up the sides of the smaller dish. Bake until the custard is set, about 1 hour. Serve warm or chilled.
Tip: Place a piece of plastic wrap directly against the surface of the custard before you refrigerate it. This prevents a skin from forming.