In a large saucepan, melt the butter and heat it to a bubble. Add flour gradually and stir constantly over medium-high heat to make a light brown roux.
Lower heat to medium and add onion, bell pepper, and celery. Cook until the vegetables soften, while stirring to keep the roux from burning.
Add the wine, stock, thyme, bay leaf, and garlic, then bring to a boil. Simmer two or three minutes, until thickened.
Add crawfish tails, tomatoes, parsley, and green onions. Simmer for about eight to ten minutes, until sauce can be picked up with a fork. Adjust seasonings with salt and Creole seasoning.
Heat a griddle or a black iron skillet over medium-high heat. Grill the tortillas until they brown in spots. Sprinkle cheese over tortillas. Put them on a pan or baking sheet three inches under the heat in a broiler until the cheese melts. Top with the hot crawfish sauce in the center (don't spread it around pizza-style). Cut into four pieces and serve hot.