Soak the rice in a bowl of water for at least 3 hours or overnight.
Make the custard. Mix together the coconut milk, eggs, sugar and vanilla extract until the sugar has dissolved. Pour the custard through a sieve into a steamer bowl until it is three-quarters full. Taking care not to burn your hand, set the custard bowl inside the steamer basket or rack, simmer for 10–15 minutes, until set around the edges, and set aside. Leave to set at room temperature for about 30 minutes.
Drain and spread the rice into the same steamer basket over a double thickness of muslin. Cover and simmer for 30–35 minutes or until the rice swells and is glistening and tender. Check and replenish the water every 10 minutes or so.
Mix together the coconut milk, measured water and sugar and set aside until the sugar has dissolved.
Spoon the rice into a bowl as soon as it is cooked. Mix in the coconut milk mixture, cover and set aside for 10 minutes. Serve the black sticky rice on a small dessert plate and spoon the custard over the top.