1. Remove and discard skin from chicken; rinse chicken and pat dry. Pull off and discard lumps of fat; with a sharp knife, separate legs and thighs.
2. In a 4- to 6-quart pan over high heat, stir bacon until browned, 2 to 3 minutes. Add onions and mushrooms and stir often until browned, 5 to 6 minutes. Add chicken, broth, wine, mustard, and thyme. Cover and bring to a boil, then reduce heat and simmer until chicken is no longer pink at the bone (cut to test), 20 to 25 minutes.
3. With a slotted spoon, remove chicken from pan and divide equally among four bowls. Increase heat to high and boil liquid and vegetables to concentrate flavors, about 5 minutes.
4. In a small bowl, blend cornstarch with 2 tablespoons water until smooth. Add to broth mixture and stir until it boils and thickens, about 1 minute. Add salt and pepper to taste. Spoon sauce equally over chicken.