In a medium bowl, stir together self-rising flour, milk, and melted butter for dumplings. Set aside.
Heat olive oil in a Dutch oven over medium heat.
Add celery, onion, carrot, and half of salt and pepper to pan. Cook, stirring occasionally, until slightly softened, about 3 minutes.
Add chicken, garlic, and remaining salt and pepper. Cook until chicken takes on just a little color, about 2 minutes per side.
Stir in broth and thyme sprigs. Bring to a simmer.
Reduce heat to medium-low; cover and simmer until chicken is cooked through, about 18 minutes.
Remove chicken from broth; shred with two forks. Return to broth, and bring back to a simmer.
Using a spoon, scoop about 1 tablespoon dumpling mixture, and drop into simmering broth.
Repeat with remaining dumpling mixture, working one at a time.
Reduce heat to medium-low; cover, and cook until dumplings are puffed a wooden pick inserted in the center comes out mostly clean, about 15 minutes.
Sprinkle servings with parsley, if desired.