Melt 2 tablespoons of the butter in a large pot over medium heat. Add the chicken pieces, onion, celery, and carrots. Cook for 6 to 8 minutes until the chicken is lightly browned and the vegetables begin to soften.
Pour in the chicken broth and stir in the thyme and black pepper. Bring the mixture to a simmer and cook for 10 minutes.
While the broth simmers, prepare the dumplings. In a mixing bowl, combine the flour, baking powder, and salt. Stir in the milk and remaining melted butter until a soft dough forms. Do not overmix.
Drop spoonfuls of the dumpling dough directly into the simmering broth. Cover the pot and cook for 12 to 15 minutes without lifting the lid.
Stir in the frozen peas during the final 3 minutes of cooking. The dumplings should be firm and cooked through when done.