In a large mixing bowl, combine ground beef, ground chicken, 1/4 cup chopped onion, 2 minced garlic cloves, 1/4 cup chopped parsley, egg, breadcrumbs, 1 teaspoon salt, and 1/2 teaspoon black pepper. Mix well until all ingredients are evenly incorporated.
Shape the meat mixture into small meatballs, about 1 inch in diameter, and set aside on a plate.
Heat olive oil in a large pot over medium heat. Add diced onion, carrots, and celery. Cook, stirring occasionally, until vegetables are softened, about 5 minutes.
Add 2 minced garlic cloves, dried thyme, dried parsley, 1/2 teaspoon salt, and 1/4 teaspoon black pepper to the pot. Cook for 1 minute until fragrant.
Pour in the chicken broth and bring to a boil.
Once the broth is boiling, carefully add the meatballs to the pot. Reduce heat to a simmer and cook for 20 minutes, until meatballs are cooked through.
While the meatballs cook, prepare the dumpling dough. In a medium bowl, whisk together flour, baking powder, and 1/2 teaspoon salt.
Add milk and melted butter to the dry ingredients and stir until just combined. Do not overmix.
Drop spoonfuls of the dumpling batter onto the simmering soup, spacing them evenly. Cover the pot with a lid and cook for 15 minutes without lifting the lid, until dumplings are puffed and cooked through.
Remove the soup from heat. Taste and adjust seasoning with salt and pepper if needed.
Serve the soup hot, ensuring each bowl has a good amount of broth, meatballs, vegetables, and dumplings.