Prepare the Stew: Begin by melting the butter in a microwave-safe measuring cup or small bowl. Once melted, whisk in the flour until the mixture is smooth and free of lumps.
Combine Stew Ingredients: In a 6- to 8-quart slow cooker, mix the butter-flour mixture with the chicken, garlic, celery, carrots, onion, lemon juice, bay leaves, tomato paste, thyme, rosemary, chicken broth, and a pinch of salt and black pepper. Stir thoroughly to ensure even distribution of ingredients. Cover and cook on the low setting for approximately 5 hours, or until the chicken is tender and the vegetables are soft.
Prepare the Dumplings: In a separate bowl, combine the flour, cornmeal, baking powder, and a small amount of salt. Pour in the buttermilk and melted butter, stirring gently until the mixture is just combined. Be cautious not to overmix to maintain the fluffiness of the dumplings.
Add Dumplings: Once the chicken and vegetables are cooked, uncover the slow cooker and use two forks to shred the chicken into bite-sized pieces. Drop the dumpling batter onto the stew using a large spoon or a ¼-cup measuring cup, creating about 8 evenly spaced dumplings on top of the stew. Cover the slow cooker and cook for an additional hour on low.
Finish and Serve: After an hour, remove the lid and continue cooking for an additional 15 minutes to firm up the tops of the dumplings. Once done, serve the dish hot, garnished with freshly chopped parsley for a touch of color and freshness.