Fill a stock pot with 4 cups of water and slide in your fish carcass along with 4 kaffir lime leaves. Turn on a medium heat. Bring to a slow simmer and turn the heat down to low for a further 25 mins, skimming off any scum from time to time. Turn the heat off, pass the contents of the pot through a sieve, and set aside.
Meanwhile, cut your whole fish into 1 inch cutlets. Bash your lemongrass with the back of your knife and cut into 2 inch pieces, discarding the end pieces. Rip single tears into the remaining kaffir lime leaves. Peel the skin off the galangal with a vegetable peeler, and slice into 2mm discs. Take the stalky hats off the chillies, and bash them like the lemongrass. Throw the lemongrass, lime leaves, coriander roots, galangal and chillies into a large pot. Pour in 3 cups from the top of your fish stock (any sediment should have settled by now) and the coconut milk. Turn on the heat to medium and bring to a slow simmer. After about 10 mins, season with fish sauce, and lime juice to taste, then add your fish cutlets and turn off the heat as soon as the fish is done. Serve immediately with a few leaves of coriander and fresh rice.