In a large saucepan, bring 6 cups of water to a boil. Remove heads and peels from shrimp. De-vein if desired.
Add peels and heads to saucepan along with celery, carrot, onion, and garlic. Boil for 45 minutes to an hour, or as time allows. The longer you boil it the more concentrated the flavors will be. Strain and set aside 2 cups of stock. Remaining stock can be reserved for another use.
To prepare grits, bring 3 cups water to a simmer in a large saucepan. Gradually whisk in grits. Turn heat to low; gently simmer until grits begin to thicken. Continue cooking, stirring often and adding water by ¼ cupfuls if too thick, until tender, about 1 hour. Stir in butter and cream. Season generously with salt and pepper. Keep warm.
Meanwhile, preheat a large saute pan over medium-high heat. Add sausage pieces and cook until browned, about 5 to 7 minutes. If dry, add 1 tablespoon butter. Add garlic, onion, and pepper and cook until softened.
Whisk in tomato paste and thyme and cook for 3 minutes. Add reserved 2 cups of shrimp stock; reduce heat to a simmer, cook for 5 to 7 minutes or until sauce has reduced to desired consistency. Season with salt and pepper to taste.
Add shrimp and cook for 2 minutes or until shrimp are just opaque and start to curl.
Divide grits among bowls. Spoon shrimp mixture over top and serve.
