Melt the butter in a pot or large skillet over medium-high heat.
Add the chopped onion, celery, bell pepper, and garlic and cook until softened, stirring occasionally.
Warm the water or chicken stock in a microwave; then pour slowly into the flour, mixing well.
When the vegetables are cooked down, add the water-flour mixture, as well as the salt and Creole seasoning, and stir.
Add the crawfish tails to the cooked vegetables and turn heat to medium-low. Simmer until the tails are cooked (about 20 minutes). After 15 minutes, add the green onions and simmer for 5 minutes more. Then remove from heat to use in the hand pies.
Preheat oven to 350 degrees and line a rimmed baking sheet with parchment paper.
Peel the casing from the boudin and put the loose boudin in a bowl.
In the middle of each dough disk, drop approx. 1 tbsp etouffee, 1 tbsp boudin and a sprinkling of cheese. Don’t pile too high, and leave plenty of space around the edges so the disk will be able to close without breaking.
Brush the edges with the beaten egg. Then fold the disk over (creating a half-moon shape), and crimp the edges with a fork to seal.
Using a fork, poke a few holes in the tops of the hand pies.
Brush the top of each hand pie with the beaten egg and place on the baking sheet.
Bake until golden brown (about 30 minutes).