In a medium Dutch oven, melt the butter over medium-high heat. Add the onion, carrots, celery and thyme; cook until the onion is translucent, about 4 minutes. Add the flour, and cook, stirring, for 1 minute. Gradually add the broth, stirring constantly, and bring to a boil. Add the chicken and return to a boil. Reduce the heat and simmer for 5 minutes. Stir in the green beans; season with salt and pepper.
MAKE THE DUMPLINGS:
In a small bowl, whisk together the flour, baking powder and salt; then whisk in the parsley. With a pastry blender or two knives, cut in the butter until the mixture resembles coarse meal. Stir in the milk with a fork just until a dough forms. Drop the batter by heaping tablespoons on top of the stew. Cover and simmer just until the dumplings are cooked through, about 12 minutes. Garnish with chopped parsley.