In a large pot or Dutch oven, brown the chicken in 1 tablespoon olive oil over medium-high heat for 5-7 minutes. Transfer to a bowl and set aside.
Add 3 tablespoons butter and the finely diced yellow onion to the pot and cook for about 3 minutes or until slightly softened. Stir in 3 tablespoons of all-purpose flour, poultry seasoning, dried thyme, and ¼ teaspoon each salt & black pepper. Cook while stirring for 2 minutes.
Gradually add 4 cups of chicken broth to the pot, stirring until smooth after each addition. Add the chicken with any juices, chopped carrots, and chopped celery. Bring to a boil, reduce heat and simmer covered for 12 minutes.
Meanwhile, to make the dumplings, in a medium bowl, whisk 1 ½ cups all-purpose flour, 2 teaspoons baking powder, and ½ teaspoon salt. Use a fork to mix in 2 tablespoons cold butter until small crumbs form. Add ½ cup milk and combine with a spoon.
Remove the lid from the soup. Stir in ½ cup frozen peas if using. Scoop small spoonfuls of dough on top of the soup to form 16 dumplings. Cover and cook for an additional 4 to 7 minutes or until the dumplings are cooked through.
Garnish with fresh parsley if desired.
Serve hot and enjoy!