Heat the butter in a large cast-iron dutch oven, over medium-high heat, until it stops foaming, then add the onions, stirring frequently until they begin to brown. Add ¼ cup of chicken stock and cook, stirring and scraping the bottom of the pot with a wooden spatula, until it's almost evaporated. About 10 minutes.
Add the balance of the vegetables (excluding the dark green parts of the onions). Cook, stirring often until softened and beginning to stick to the bottom of the dutch oven. Add ¼ cup of chicken stock. Cook, stirring and scraping the bottom of the pot, until evaporated, then add ½ cup of stock and cook again, stirring often, until evaporated. About 10 minutes.
Add the garlic, flour, and spices, and cook, stirring constantly until all of the ingredients are well coated. About 1 minute.
Add the remainder of the chicken stock, stirring and scraping the bottom of the pot with a sturdy wooden spatula, to remove any stuck-on bits, then add the crawfish, and stir again. Bring to a simmer, then reduce the heat to medium-low and cook, uncovered, for an additional 20 minutes, or until the gravy is the consistency you like.
Taste and adjust for salt. Serve over steamed rice, garnished with the dark parts of the green onions.
