Preheat slow cooker on low setting for 20 minutes. Pat chicken legs dry and season them well with salt and pepper. Heat 2 tablespoons of olive oil in a large skillet over medium heat. When the oil is shimmering, add chicken legs and sear them on one side for 5 minutes. Then flip them over and sear for another 5 minutes, until nicely browned on both sides.
Pour broth into slow cooker and add onion, half of celery, mushrooms, and bay leaves. Add chicken, cover, and cook for 4 hours.
Once chicken has cooked for about 3½ hours, prepare dumpling dough. In a medium bowl, combine flour, buttermilk, 1 teaspoon salt, and baking powder and mix with a fork until just combined (a few lumps are okay). When chicken has cooked 4 hours, add dumplings by dropping heaping spoonfuls (golf ball-size) directly into slow cooker. Cover with lid and cook for 30 minutes, then flip dumplings over and cook for 30 minutes more.
To finish the dish, place a medium skillet over medium heat and add remaining 1 tablespoon olive oil. When oil begins to shimmer, add diced carrots, remaining celery, and a pinch of salt. Sweat vegetables for 3 minutes, then remove skillet from heat and add thyme and parsley. Fold this mixture into slow cooker and serve.