In a large saucepan, combine the half and half, chicken stock, and water and bring to a simmer. Whisk in the grits and season lightly with salt and white pepper. Bring to a low boil, adjust the heat to a steady, gentle simmer, and cook the grits (stirring frequently), until they are very tender and thickened, 45 minutes to 1 hour. To prevent a skin forming on the top of the surface, keep the pan partially covered. When the grits are tender and creamy, stir in the Parmesan cheese and then adjust the seasoning with salt and white pepper.  Cover the pot to keep warm and set aside while you cook the crawfish.

In a 3 quart saucepan or in a medium saute pan, melt 3 tablespoons of the butter. Add the onion, celery, and the red and green bell peppers and cook until vegetables are softened, 4 to 6 minutes. Stir in the garlic and cook for 2 minutes. Sprinkle the flour over the vegetables and season lightly with salt and cayenne pepper. Cook, stirring, for 2 minutes. Stir in the wine, stock, and cream and add the thyme sprigs. Cook, stirring, until sauce is thickened and flavorful and any floury taste is gone, 5 to 10 minutes. Add the crawfish and cook just until they are heated through. Season a final time with salt and cayenne pepper to taste, and remove from the heat. Remove the thyme sprigs, and stir in the remaining 2 tablespoons of butter and the parsley.

Serve the grits in shallow bowls, with the crawfish spooned over the top. Garnish with additional parsley if desired.