Melt 2 tablespoons of the butter in a Dutch oven over medium-high. Add carrots, celery, and onion; cook, stirring occasionally, until onions are tender, 5 to 6 minutes. Add 2 tablespoons of the flour; cook, stirring occasionally, 1 to 2 minutes. Add broth, bay leaf, and 1/2 teaspoon of the salt; bring to a boil. Cook, stirring occasionally, until thickened, 4 to 5 minutes. Stir in chicken. Remove Dutch oven from heat.

Whisk together cornmeal, baking powder, baking soda, and remaining 1/2 cup flour and 1/4 teaspoon salt in a medium bowl. Cut remaining 2 tablespoons butter into flour mixture with a pastry blender or fork until mixture resembles coarse meal. Add buttermilk, parsley, and tarragon, stirring just until dough is moistened.

Heat Dutch oven over medium. Drop dumpling dough by tablespoonfuls 1/2 to 1 inch apart into chicken mixture. Cover and cook until dumplings are done and dry to the touch, 15 to 20 minutes. Serve immediately.