There are two staple soups in Thai cuisine: tom yam is a hot spicy clear soup with elements of sweet and sour flavors added. Tom kha is a milder soup with coconut milk and galangal (kha) dominating rather than the fiery prik (chili) of the tom yam. Because it is milder tom khas are often made with chicken or pork, but most common in Thailand are varieties using seafood (especially shrimp, squid, or fish such as red snapper or catfish) or vegetables (especially medleys of mushrooms, tom kha hed). The catfish can be "crisped" by quickly, and briefly, deep frying it in very hot oil, but this variation is based on simply poaching the fish in the soup.
For 4 Person(s)
In Thailand the fish is cleaned and then poached whole (with the head), then removed from the soup and cut into bite sized pieces which are returned to the soup for serving. The method here is a little simpler, in that it doesn't involve handling the hot fish.
Thais eat the galangal, which is cut into thin matchstick pieces. However it has been noticed that many Americans prefer to discard the galangal and so it may be wiser to leave the galangal in thin slices. Similarly the lemongrass is eaten, but you may prefer to cut it into 2" lengths, and crush them with a mallet. These may then be discarded by the diner.
Bring the stock to a simmer. Add the galangal, lemongrass, coriander, chilis, fish sauce and lime juice, and bring back to the simmer. Clean the fish and cut it into 1" steaks, then divide them, removing the bones. Add the fish to the soup, and the coconut milk and bring back to a very gentle simmer, and poach the fish for 3-4 minutes (until just cooked).