Fill a bucket or your kitchen sink with two or three gallons of cold water with about a half-cup of salt dissolved in it. Dump the crawfish in; the salted water will purge them. Rinse with two or three changes of water until the water is only slightly dirty. Some cooks say that has no effect on anything, but it seems to me worth doing.
Bring a large stockpot with five gallons of water to a boil. Add all the other ingredients except the crawfish and potatoes and return to a boil. Let it cook for fifteen minutes.
Add the crawfish and the potatoes. Return to a boil, making sure there’s enough water to completely cover the crawfish.
After eight minutes, remove the biggest crawfish you see and open it up to make sure the tail meat is firm and opaque. If not, give it another couple of minutes of boiling, but no more than that. If the crawfish are indeed done, turn off the heat and let the crawfish steep for 20-30 minutes. Remove the potatoes when they're tender. Take the crawfish out when they've absorbed the seasonings to the degree you like.
At this point, we commence the peeling and eating process which, if you haven’t learned it, you’re better off picking up the technique from a friend than reading about it. The potatoes are a side dish. Discard everything else. Rinse, freeze, and save the crawfish shells for making bisque or etouffee or sauces.