Preheat the oven to 400°F (200°C). Lightly grease an 8-by-8-inch baking pan with cooking spray or butter.
In a medium bowl, whisk together the flour, cornmeal, sugar, baking powder, and salt.
In a separate bowl, whisk together the milk, vegetable oil, and egg. Pour the wet ingredients into the dry ingredients and stir just until moist and nearly lump-free.
Transfer the batter to the prepared pan. Bake for 20–25 minutes, or until golden and a toothpick inserted into the center comes out clean. Cool completely, then crumble the cornbread.
Bring several inches of water to a boil in a large Dutch oven. Working in batches, boil the bell pepper halves for about 7 minutes. Drain well and allow them to cool.
Place the shrimp in a colander or bowl and toss with 1 teaspoon Cajun seasoning. Set aside while preparing the vegetables.
Reduce the oven temperature to 375°F (190°C).
Melt 1 tablespoon butter with the olive oil in the Dutch oven over medium-high heat. Add the diced onion and cook for about 5 minutes.
Add the diced bell pepper, celery, and white portions of the green onions. Sauté for another 5 minutes, or until softened. Stir in the garlic and thyme and cook for 1 minute.
Add the seasoned shrimp and cook briefly, stirring, until the pieces are just beginning to turn pink. Remove from the heat to avoid overcooking.
Add the crumbled cornbread to the shrimp and vegetable mixture. Fold in 1 cup chicken broth, beaten eggs, sage, Tabasco, green portions of the onions, parsley, and the remaining teaspoon of Cajun seasoning.
If the stuffing seems dry, gradually add more chicken broth, about 2 tablespoons at a time, until moist but not soggy.
Spoon the filling generously into each pepper half, pressing it gently into the corners before mounding the stuffing on top. These Cajun Stuffed Bell Peppers with Shrimp should be generously filled.
Arrange the peppers stuffing-side up in a baking dish. Divide the remaining butter into small pieces and dot it over the stuffing.
Bake uncovered for approximately 25 minutes, until the filling is hot and lightly golden on top. Garnish with additional parsley or fresh thyme before serving.