Melt the butter in a large saute pan over medium high heat. Add the onions, celery, and bell peppers and sauté until the vegetables are wilted, about 10 to 12 minutes. After 5 minutes, stir in the tomato paste and continue to sauté. Once vegetables are softened, add the garlic, and bay leaves and reduce the heat to medium. Cook the crawfish for 10 minutes, stirring occasionally.
Whisk the flour into the stock and pour into the pan. Season the mixture with a pinch of salt and cayenne. Cook while stirring until the mixture thickens, about 4 minutes.
Stir in the parsley and green onions and continue cooking for 2 minutes. Serve over grits with a poached egg.
For the Grits
Heat oven to 350 degrees.
Add chicken stock, salt, 2 tablespoons butter and grits to a 3 quart baking dish and stir to combine. Bake for 1 hour, stirring after 30 minutes.
Remove from oven and whisk in the heavy cream, butter, and goat cheese to finish and loosen the grits. Taste and adjust seasoning as necessary.
For the Poached Eggs
Fill a large high sided skillet halfway with water. Heat over medium heat and stir in salt and vinegar.
Crack an individual egg into a ramekin. Use spoon to stir water to create a whirlpool then drop egg right into the center of the pan. Cook for about 4-5 minute then remove using a slotted spoon. Drain off on a plate lined with paper towel. Serve immediately right on top of the crawfish and grits.