My idea is to have two separate mulch pit/swale/pond the 1st one catches alot of the grease and such.
Dont worry the microbes will get rid of it no problem water would them flow form this to another mulch pit/swale.
This is in my opinion the best way to do it. It has a filter system, it has a overflow.
You should not under any circumstances let greywater sit for more than 24 hours, why?
It has ecoli form the chicken sink, and shitty hand/spit in bathroom, bodily excrement/fluid from the bathroom, etc.
The grey water is also high in toxins and salt so I would not use it to water my vegetable, only fruits a few feet off the ground.
Leaves are usually high in toxins/salt(mineral) they are close to the ground (ecoli and others).
You could also plant vines and then chop and drop using it as green manure.
You also want the salts (laundry soaps, etc) to soak deep into the ground and to be evenly distributed so dont worry if the vegetables dont get the alot of grey water.
Find tree that use up alot of nitro and phosphate (tomatoes) vs adding more salt to your soil.
In theory you could send the water to a pond, have t bloom with algae using up the nutrients, and other microbes in the pond will ou compete the bad bugs.
You would have to put tilapia in the pond to eat the algae. then a aquaponic system to clear the water of nitrates and then feed the vegetables to chickens.
And the chicken/duck poop could also fall back into the pond thus fertilizing it.
http://gardenpool.org