In a large heavy-bottomed pot or Dutch oven, heat a drizzle of oil over medium heat. Add the diced onion, carrots, and celery. Cook for about 5 to 7 minutes until the onions are translucent and the vegetables begin to soften slightly.
Pour in the chicken broth and add the dried thyme, salt, and pepper. Place the chicken pieces into the pot. Bring the liquid to a gentle boil, then reduce the heat to low. Cover the pot and simmer for 20 minutes, or until the chicken is fully cooked through and easy to shred.
Remove the chicken from the pot and place it on a cutting board. Use two forks to shred the chicken into bite-sized pieces. Return the shredded chicken to the simmering broth.
In a medium mixing bowl, whisk together the flour, baking powder, and 1/2 teaspoon of salt. Make a well in the center and pour in the milk and melted butter. Stir with a fork just until the dough comes together. Be careful not to overmix, as this can make the dumplings tough.
Using a spoon or a small cookie scoop, drop rounded tablespoons of dough directly onto the simmering broth. Space them out as much as possible. Cover the pot tightly with a lid and cook for 15 minutes without lifting the lid. This allows the dumplings to steam and rise properly.
Check the dumplings by inserting a toothpick into one; it should come out clean. Gently stir the pot to incorporate everything, taste the broth, and add more salt or pepper if needed. Serve immediately while hot.