1. Preheat oven to 500°.
2. Weigh or lightly spoon self-rising flour into dry measuring cups; level with a knife. Place self-rising flour in a bowl. Cut in butter with a pastry blender or 2 knives until mixture resembles coarse meal. Add buttermilk; toss with a fork until moist. Drop dough by tablespoonfuls onto a foil-lined baking sheet coated with cooking spray to make 32 biscuits. Bake at 500° for 12 minutes or until edges are browned. Cool on wire racks. Reduce oven temperature to 350°.
3. Heat a large Dutch oven over medium-high heat. Add 1 tablespoon oil to pan; swirl to coat. Sprinkle chicken with 1/2 teaspoon salt and 1/2 teaspoon pepper. Add half of chicken to pan; cook 5 minutes on each side or until browned. Remove from pan. Repeat procedure with remaining chicken. Remove from pan.
4. Add carrot and next 5 ingredients (through shallots) to pan; cook 5 minutes or until lightly browned, stirring occasionally. Add wine; cook until liquid is reduced to 1/4 cup (about 2 minutes). Add stock; bring to a boil. Add browned chicken. Cover and place in oven; bake at 350° for 45 minutes or until chicken is very tender. Remove from oven. Remove chicken from pan, and let stand 5 minutes.
5. Remove chicken from bones; discard bones. Strain liquid over a bowl; reserve solids. Combine 1/2 cup cooking liquid and all-purpose flour in a small bowl, stirring with a whisk until smooth. Add flour mixture and remaining cooking liquid to pan. Bring to a boil; cook 3 minutes or until slightly thickened.
6. Heat a large skillet over medium-high heat. Add remaining 1 tablespoon oil; swirl to coat. Add Brussels sprouts, cut sides down, to pan; cook 3 minutes or until lightly browned. Add remaining 1/2 teaspoon salt, remaining 1/4 teaspoon pepper, and mushrooms to pan; sauté 5 minutes.
7. Combine chicken, carrot mixture, cooking liquid, and mushroom mixture in a 13 x 9-inch glass or ceramic baking dish. Arrange biscuits evenly over chicken mixture. Bake at 350° for 20 minutes or until bubbly.