Preheat oven to 325°F. Cook bacon and oil in a Dutch oven over medium-high until bacon is crisp. Transfer to a bowl. Add chicken, salt, and pepper to pot; cook until browned. Transfer chicken to bowl with bacon.
Add mushrooms to pot; cook 5 minutes. Transfer to bowl with meat. Add carrot, onion, garlic, and bay leaves to pot; cook until softened. Add vinegar; cook until reduced, 2 minutes. Return chicken mixture to pot. Remove from heat.
Melt butter in a medium saucepan over medium-high until foamy. Add flour; cook, stirring constantly, 1 minute. Add milk; cook, whisking often, until thickened. Add broth and mustard; cook, whisking constantly, until thickened. Pour over chicken mixture, cover, and cook at 325°F 1 hour. Discard bay leaves. Stir in parsley; serve over rice.