Pat the chicken thighs dry and season them lightly with salt and pepper.
In a large pot, melt the butter over medium heat. Add the onion, carrot, and celery, and sauté until they begin to soften and develop a little color, about 5 minutes.
Add minced garlic and thyme; cook for another minute until fragrant.
Pour in the chicken stock and bring to a gentle simmer. Add the seasoned chicken thighs to the pot.
Simmer, uncovered, for about 25 minutes, or until the chicken is cooked through and tender. Skim off any foam that forms on the surface.
Remove the chicken to rest briefly, then shred the meat, discarding the bones. Return shredded chicken to the pot.
Make the dumpling batter: in a bowl, whisk together the flour, milk, a pinch of salt, and a splash of water if needed to form a thick, scoopable batter.
Drop spoonfuls of the batter onto the simmering soup. Cover the pot and cook for 12 to 15 minutes, or until the dumplings are fluffy and cooked through.
Check for seasoning, adjust with salt or a squeeze of lemon juice if desired, and finish with a final grind of black pepper.