Place the chicken in a pot with the chicken broth, bay leaves, and a little of the kosher salt. Bring to a gentle simmer and cook for about 15–20 minutes, or until the chicken reaches 165°F internally. Remove the chicken, shred or chop it, and reserve it while keeping the broth.

Melt the butter in a large Dutch oven or heavy pot. Add the onion, carrots, and celery, then cook for 5–7 minutes until the vegetables begin to soften. Add the garlic and cook for about 30 seconds.

Stir the flour into the softened vegetables and butter. Cook for 1–2 minutes, stirring often, to remove the raw flour taste. Gradually whisk in the reserved broth, then add the milk, thyme, black pepper, and remaining salt.

Simmer the soup for about 10 minutes, stirring occasionally, until the broth is lightly thickened. Return the shredded chicken to the pot and add the frozen peas if using.

For the dumplings, whisk together the flour, baking powder, and salt in a bowl. Cut in the cold butter or shortening until the mixture resembles coarse crumbs, then stir in the milk just until a soft, slightly sticky dough forms. Add 1–2 tablespoons more milk only if needed.

Drop the dumpling dough by rounded tablespoonfuls over the surface of the gently simmering soup. Cover the pot tightly and cook for 15 minutes without lifting the lid. Remove the lid and check that the dumplings are fluffy and cooked through in the center; cook 2–5 minutes longer if necessary.

Remove the bay leaves, taste the soup, and add more salt if needed. Garnish with chopped fresh parsley if desired, then serve hot.