Sauté the base. In a large pot or Dutch oven, heat butter and olive oil over medium heat.
Add onion, carrots, and celery with a pinch of salt. Cook 5–7 minutes until softened. Stir in garlic, thyme, and bay leaf for 30 seconds until fragrant.
Build the roux. Sprinkle the flour over the vegetables.
Stir for 1–2 minutes to coat and lightly toast the flour. This step thickens the soup and adds flavor.
Add the broth. Slowly whisk in the chicken broth, scraping up any bits on the bottom. Bring to a gentle simmer and cook 10 minutes, stirring occasionally, until slightly thickened.
Add chicken and peas. Stir in the shredded chicken and peas.
Season with salt and pepper. Keep at a gentle simmer while you make the dumplings.
Make the dumpling dough. In a bowl, whisk flour, baking powder, salt, and garlic powder. Cut in the cold butter with your fingers or a pastry cutter until it looks like coarse crumbs.
Stir in milk and herbs until a soft, sticky dough forms. If it’s too dry, add 1 tablespoon milk at a time. Do not overmix.
Drop the dumplings. Reduce the soup to a low simmer.
Using a spoon or small scoop, drop heaping tablespoon-sized portions of dough onto the surface. Leave a little space between each so they can expand.
Steam the dumplings. Cover the pot with a tight lid and cook for 12–15 minutes without lifting the lid. The steam cooks the dumplings through and keeps them tender.
They should be puffed and cooked in the center.
Finish the soup. Stir in the cream and fresh parsley. Taste and adjust seasoning with more salt and pepper. If the soup is too thick, add a splash of broth.
If too thin, simmer uncovered for 3–5 minutes.
Serve. Ladle into bowls and garnish with extra parsley and black pepper. Serve hot.