Soften the vegetables. Melt the butter in a wide 6- to 7-quart Dutch oven over medium heat. Add the onion, carrots, and celery. Cook for 7 to 9 minutes, stirring often, until beginning to soften.
Add the seasonings and flour. Stir in the garlic, thyme, and sage for 30 seconds. Sprinkle in the flour and cook, stirring, for 1 minute.
Add the broth. Gradually stir in the broth, scraping the bottom of the pot. Bring to a gentle simmer and cook uncovered for 5 to 7 minutes, until the vegetables are nearly tender and the broth lightly coats a spoon.
Add the chicken. Stir in the chicken, cream, and optional peas. Simmer gently for 3 to 5 minutes. Taste before adding salt and pepper. Add extra broth if the base looks thick like gravy.
Make the dumpling dough. Whisk the dumpling flour, baking powder, and salt in a medium bowl. Stir the melted butter into the buttermilk, then add to the flour mixture with the optional parsley. Fold just until no dry flour remains. Add extra buttermilk by the tablespoon only if the dough is dry.
Add the dumplings. With the base gently simmering, drop the dough into 12 similar mounds, about 3 tablespoons each, leaving small gaps. Keep the dumplings on the surface and do not stir.
Cover and cook. Maintain a gentle simmer for 18 minutes without lifting the lid. Cut open a large dumpling. If its center is wet or doughy, cover and cook for another 3 to 5 minutes, then check again. Continue until the centers are cooked through.
Rest and serve. Remove from the heat and rest uncovered for 5 minutes. Serve with fresh parsley and additional black pepper.