In a large pot, combine 2 cups of chicken broth and 2 cups of water. Bring it to a boil over medium-high heat. Once boiling, gradually whisk in 1 cup of old-fashioned grits along with 1/2 teaspoon of salt. Reduce the heat to low, cover, and let it simmer for about 20-25 minutes, stirring occasionally until the grits are thick and creamy.
Once the grits are cooked, remove them from heat and stir in 3/4 cup of freshly grated Parmesan cheese, 1/4 cup of heavy cream, and 1/2 teaspoon of pepper. Mix well until the cheese is melted and creamy. Keep the grits warm while you prepare the shrimp.
In a large skillet, heat 2 tablespoons of butter and 1 tablespoon of olive oil over medium heat. Add 1/2 medium diced onion and 1/2 diced green bell pepper. Sauté until the vegetables are tender, about 5 minutes. Next, add 3 diced links of Andouille sausage and cook for another 3-4 minutes until the sausage is browned.
Stir in 3 minced garlic cloves and cook until fragrant, about 1 minute. Sprinkle in 2 tablespoons of all-purpose flour, stirring constantly for about 1 minute to create a roux.
Slowly pour in 1 can of chicken broth and 1 can of undrained diced tomatoes. Mix well. Season with 2 teaspoons of Cajun or Creole seasoning and 1/2 teaspoon of cayenne pepper. Let the mixture simmer for about 5 minutes.
Add 1 pound of peeled and deveined shrimp to the skillet. Cook for approximately 3-5 minutes, or until the shrimp are pink and opaque. Stir in 1/3 cup of heavy cream and let it simmer for another 2 minutes.
To serve, spoon a generous portion of the creamy Parmesan grits onto a plate and top with the smothered shrimp mixture. Garnish with chopped parsley and sliced green onions for a fresh touch.