Increase the oven temperature to 400?F. Grease a 10 x 15-inch baking dish or two 9-inch square baking dishes with the butter.
Combine the Chicken stock, cream, milk, eggs, and 1 tablespoon Essence in a large bowl and whisk to blend. Add the dried cornbread and stir to mix, breaking up the pieces with a wooden spoon. Cover and refrigerate for 1 hour.
Meanwhile, heat a large skillet or medium pot over high heat. Add the andouille and cook until browned, about 5 minutes. Add the vegetable oil and reduce the heat to medium-high. Add the onions, celery, bell peppers, remaining 2 teaspoons Essence, the salt, black pepper, and cayenne and stir to mix. Cook the vegetables, stirring occasionally, until very soft and golden, about 15 minutes. Add the garlic and thyme and cook until fragrant, about 30 seconds. Remove from the heat and add to the cornbread mixture. Stir to mix well. Pour into the prepared baking dish and bake until golden brown, 35 to 40 minutes.
Remove from the oven. Cool if using as stuffing or serve warm as a side dish.
Makes 10 to 12 servings