In a large pot or Dutch oven, heat the olive oil over medium heat. Add the onions, carrots, celery, and garlic. Sauté for about 5-7 minutes until the vegetables are softened.
Add the chicken thighs to the pot, followed by the chicken broth, thyme, salt, and pepper. Bring to a boil, then reduce the heat to low, cover, and let it simmer for about 20 minutes until the chicken is cooked through.
While the chicken is cooking, prepare the gluten-free dumplings. In a mixing bowl, combine the gluten-free flour, baking powder, and salt.
In another bowl, whisk together the milk, beaten eggs, and melted butter. Pour the wet ingredients into the dry ingredients and stir until just combined. The mixture should be thick but should drop from a spoon.
Remove the cooked chicken from the pot and shred it using two forks, then return the shredded chicken to the pot.
Bring the broth back to a gentle simmer. Using a spoon, drop spoonfuls of the dumpling batter into the simmering broth. Do not overcrowd; these will expand while cooking.
Cover the pot and let the dumplings cook for about 15 minutes without lifting the lid. They should be fluffy and cooked through.
Taste the broth and adjust seasoning if needed. Serve the chicken and dumplings hot in bowls.