1. Peel and quarter the onion. Peel and roughly chop the garlic. Slice the carrots and celery.
2. Place the onion and garlic in the bowl fitted with the ultrablade knife. Launch at speed 10 for 10 seconds. Replace the ultrablade knife with the whisk. Add 1/4 cup butter and launch the P1 simmer program. Add 1/3 cup flour, chicken broth, 1/2 tsp. salt and pepper. Mix at speed 7 for 1 minute.
3. Replace the whisk with the mixer. Add carrots and celery. Launch the P2 simmer program at 210°F for 15 minutes or until carrots are tender. Stir in the chicken. Relaunch the P2 simmer program at 210°F for another 5 minutes. Remove the mixer.
4. To make dumplings, stir together 1 cup flour, 1/4 tsp. salt and baking soda in a bowl. Combine buttermilk, 2 Tbsp. melted butter and egg. Add this mixture to the flour and stir with a fork just until combined. Drop dough from a spoon directly onto the chicken mixture. Launch the P2 simmer program at 210°F for 15-20 minutes or until dumplings are puffed and no longer doughy.