Melt butter in a large pot over medium heat. Add chopped onion, carrots, and celery—cook for about 3 minutes until softened. Pour in chicken broth and add the bay leaf, thyme, salt, and pepper. Bring to a boil, then add your chicken breast whole.
Reduce heat to medium and simmer the chicken for 15 minutes until cooked through. Remove chicken and shred it using two forks—mine always sticks a little and that's totally fine. Return shredded chicken to the pot and keep the broth simmering.
While the chicken cooks, make your dumpling dough. Whisk together flour, baking powder, salt, and pepper in a bowl. In another bowl, beat the egg and milk together until combined. You'll hear it bubble slightly as you mix.
Pour the wet ingredients into the dry ingredients and stir until just combined—don't overmix or your dumplings get tough. The dough should be thick but still slightly sticky, kinda like thick pancake batter.
Drop spoonfuls of dough directly into the simmering broth, spacing them out so they don't stick together. Use a regular tablespoon and just plop it in—they'll expand as they cook. Don't stir for the first 2 minutes after dropping them.
Gently stir after 2 minutes, then simmer for another 8 minutes until the dumplings are cooked through. When your kitchen smells like warm, savory broth with that buttery undertone, they're done. Taste and adjust seasoning as needed—a little extra salt makes all the difference.
Remove the bay leaf and ladle into bowls while everything's piping hot. Serve immediately and watch everyone come back for thirds. This is the kind of meal that gets saved and pinned for future weeknights.