Preheat Oven: Set your oven to 400°F.
Sauté Vegetables: In a Dutch oven, melt butter over medium-high heat. Add the carrots and onion, sautéing for about 5 minutes. Add mushrooms and cook for another 5 minutes or until tender. Stir in the garlic and cook for an additional 2 minutes.
Add Wine and Flour: Pour in the wine and cook for 2 minutes. Sprinkle in AP flour, stirring constantly for about 3 minutes.
Make the Sauce: Slowly add the broth, stirring continuously. Bring to a boil, stirring for about 2-4 minutes until thickened. Stir in the cream along with the next 5 ingredients, then add the chicken. Season with salt and pepper to taste. Cover and remove from heat.
Prepare Biscuit Dough: In a medium bowl, whisk together self-rising flour and sugar. Make a well, then in the center mix the buttermilk and melted butter, then mix until the dough comes together and starts to pull away from the sides of the bowl.
Assemble and Bake: Heat up the chicken mixture if it cooled off any, stirring constantly for about 2 minutes until hot and bubbly. Spoon the mixture into a buttered 3-quart baking dish. Drop biscuit dough by 1/4 cup portions, spacing them ½ inch apart on top of the chicken mixture.
Bake: Bake at 400°F for 30-35 minutes, until the top is golden and the dish is bubbling.