For those wondering if this is bugged for you:
It's not, you just have a specific task such as buy salted fish or taste the soup.
Check the actual message/task that is on top of the questlog on your screen, under the event timer and complete it, this should give you your credit towards the quest.
Just randomly killing things and kicking stuff back into the soup will NOT give you credit. You can kick the fish all you want, if the chef has given you no task or another task nothing will happen to complete your quest.
Again, to complete the quest if it is "bugged":
1. Check if you have a specific task under the event timer on top of your questlog. Such as defend the soup, stir the pot, taste the soup, buy the fish etc. if not go to step 3
2. Do what the chef tells you specifically via the questlog task , not what he shouts to everyone. this does not matter to you.
3. Stay next to the pot until the prompt "wait for your orders from the chef" on your questlog on screen changes into a new command
3. Repeat