First preheat your oven to 350 degrees Fahrenheit (175 degrees Celsius).
To prepare your mise en place, cut the dark green part off of the leeks. Slice in half and wash thoroughly. Slice the leeks into 1/4" pieces. Cut the carrots and celery into medium dice. Note: You should have roughly 1 to 1 1/2 cups of each. Mince the garlic.
Measure out the butter and flour. Note: The amount of flour used will produce a relatively thick sauce; however, if you want the sauce to be really thick, you can use up to 18 tbsp of flour. The roux will just be very thick.
Gather the bay leaves and thyme. Set aside.
Once the broth is cool, you can begin to make the sauce.
To make the sauce, heat a heavy-bottomed pot over medium to medium-low heat. Add the butter, followed by the celery, carrots, leeks and a pinch of salt. Sweat until the leeks are translucent, about 10 minutes or so. Once softened, add the garlic and cook until fragrant.
Singer with the flour to make a roux. Stir to combine. Temper in the cold broth a bit at a time. Bring the sauce to a simmer. Add the bay leaves and thyme and let cook for about 10 to 15 minutes to cook off any starchy flavor. Stir often to prevent scorching. Season to taste with salt and pepper.
Add the reserved chicken pieces and continue to simmer for about 10 minutes.
While the sauce is simmering, prepare the dumpling mixture (see Step 3).