- Preheat oven to 400°F. Bring broth and 1 teaspoon of the salt to a boil in a medium saucepan over high. Whisk in grits; cook, whisking often, until smooth, about 20 minutes. Add in cheese, tomato paste, and egg. Stir until blended. Set aside until ready to use.
- While grits cook, heat oil in a large skillet over medium-high. Add onion, peppers, and carrots. Cook, stirring occasionally, until vegetables are softened, 7 to 8 minutes. Add garlic, thyme, black pepper and 1/4 cup of the scallions; cook, stirring occasionally, 2 minutes.
- Stir in tomatoes and wine. Bring to a boil over high; reduce heat to medium-low, and simmer, stirring occasionally, until slightly thickened, 4 to 5 minutes.
- Remove from heat and stir in shrimp and remaining 2 teaspoons salt. (Do not cook shrimp all the way through, simply stir to blend with ingredients)
- Spread grits mixture into a 13- x 9-inch casserole dish. Top evenly with shrimp mixture.
- Cover with aluminum foil. Bake in preheated oven until casserole is bubbly and shrimp are cooked through, about 18 minutes.
- Remove from oven; let stand 5 minutes. Sprinkle with remaining 1/4 cup scallions.