Grab a large pot or Dutch oven – the bigger the better! Melt the butter over medium heat.
Add your diced onion, carrots, and celery. Sauté them for about 8-10 minutes, stirring occasionally, until they start to soften and the onion is translucent. You're building the flavor base here, so don't rush this step!
Stir in the minced garlic and cook for just one minute more until fragrant. Be careful not to burn it.
Sprinkle the flour over the vegetables and stir constantly for another minute or two. This cooks the flour and helps prevent a raw flour taste in your gravy. You've just made a quick roux!
Slowly whisk in the evaporated milk (or half and half) until smooth, then gradually whisk in the chicken stock. Make sure to scrape the bottom of the pot to get any bits of flavor up. Bring the mixture to a gentle simmer, stirring often.
Add the shredded cooked chicken, thyme (if using), and black pepper to the pot. Stir everything together. Taste the broth now and add salt as needed. Remember, you can always add more salt, but you can't take it away!
Reduce the heat to low so the soup is just barely simmering while you make the dumplings. Keep it warm.
Now, for the dumplings! In a medium bowl, whisk together the 2 cups of flour, baking powder, pepper, salt, and thyme (if using).
In a separate small bowl, whisk together the milk and melted butter.
Pour the wet ingredients into the dry ingredients and stir with a spoon or spatula just until combined. Don't overmix! A few lumps are totally fine. The dough will be sticky.
Increase the heat on your soup back to a gentle simmer – you need it hot enough to cook the dumplings, but not boiling violently. Carefully drop spoonfuls of the dumpling dough onto the surface of the simmering soup. You can use two spoons: one to scoop the dough and one to gently push it off into the pot. They don't need to be perfectly shaped!
Cover the pot tightly with a lid. Reduce the heat back to low and let the dumplings cook undisturbed for 15-20 minutes. Resist the urge to peek! Keeping the lid on traps the steam, which is essential for cooking the dumplings through and making them light and fluffy.
After 15-20 minutes, check a dumpling to make sure it's cooked through (it should be light and airy inside, no raw dough). The soup will have thickened beautifully. Give it a final taste and adjust salt and pepper if needed.
Serve hot, garnished with a little extra fresh thyme or parsley if you like. Enjoy that cozy bowl!