This is pure comfort food at its finest. Tender chunks of chicken and vegetables come together in a creamy, richly-flavored sauce with pillowy dumplings baked right on top. It's hearty but not too heavy and comes together from scratch in about an hour.
For the chicken: Heat the oil in a large Dutch oven or heavy-bottomed pot with a lid over medium-high heat until shimmering. Add the carrots, celery, onion, 1/2 teaspoon salt and about 10 grinds black pepper and cook, stirring occasionally, until the vegetables just begin to soften but don't take on any color, 4 to 5 minutes.
Stir in the garlic and thyme and cook until fragrant, about 1 minute. Add the wine and simmer until completely evaporated, 4 to 5 minutes. Stir in the butter until completely melted. Sprinkle the flour over the vegetables and stir until it is completely incorporated and a sandy paste forms. Pour in the milk and stir until the flour has been smoothly incorporated into the liquid.
Add the chicken and the chicken stock to the pot and stir to combine. Bring the mixture to a boil then lower the heat to medium-low and simmer until the chicken is cooked through, 15 to 20 minutes. Stir in the parsley and peas and remove from the heat.
For the dumplings: Whisk together the flour, baking powder, garlic salt, 1/2 teaspoon salt and about 10 grinds black pepper in a medium bowl. Stir in the melted butter until just combined, then add the milk and stir until a ball of dough forms.
Use a small ice cream scoop or measuring cup to scoop 10 balls of dough into the pot (about 2 tablespoons each). Cover the pot and simmer over medium-low until the dumplings have puffed up and a toothpick inserted into the center comes out clean, 10 to 15 minutes. Divide among bowls and sprinkle with more parsley.
Because this is the "best" chicken and dumplings, we prefer the robust flavor of homemade chicken stock. If you don't have any, you can substitute low-sodium chicken broth.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)