Heat 3 tablespoons butter in a large Dutch oven or wide pot over medium-high heat. Add the carrots, celery, parsnip, bay leaf, 1/4 teaspoon salt and a few grinds of pepper. Cook, stirring occasionally, until the vegetables are softened, about 5 minutes. Add 3 tablespoons flour; stir until toasted, about 1 minute. Gradually pour in the chicken broth, stirring. Bring to a boil, then reduce the heat and simmer until slightly thickened and the vegetables are tender, 10 to 12 minutes. Season with salt and pepper.
Meanwhile, whisk the remaining 1 1/2 cups flour with the baking powder and 3/4 teaspoon salt in a medium bowl. Cut the remaining 4 tablespoons butter into small cubes and rub into the flour mixture with your fingers until pea-size pieces form. Stir in the buttermilk and chopped herbs with a fork until combined. Divide the dough into twelve 1 1/2-inch balls.
Stir the chicken into the pot. Arrange the dough balls on top. Cover and cook over medium-low heat until the dumplings are firm to the touch and cooked through, 12 to 15 minutes. Top with more dill and chives.