Sauté the Veggies: Heat a large pot over medium heat. Add olive oil. Once hot, add the celery, carrots, and onion. Sauté for about 3 minutes until they start to soften. Add the garlic and sauté for another 30 seconds, just until fragrant. Then, add the butter and flour. Whisk well for 1 minute until it forms a roux, thickening the mixture.
Add Broth and Seasonings: Gradually stir in the chicken broth and bouillon, scraping up any brown bits from the bottom of the pot. This adds great flavor! Then add thyme, sage, rosemary, salt, and pepper. Allow this to simmer for about 5 minutes.
Add the Chicken: Introduce the chicken into the pot. Bring the mixture to a boil, then cover with a lid, reduce heat, and let it simmer for about 10 minutes or until the chicken is cooked through. Once done, remove the chicken and place it on a cutting board to cool briefly.
Stir in Milk and Peas: While the chicken cools, stir in the evaporated milk and peas into the pot. This will add creaminess and texture to the soup.
Make Dumplings: In a separate bowl, combine flour, baking powder, and salt. Make a well in the center and add milk, sour cream, and melted butter. Stir until just combined. Avoid over mixing to keep the dumplings tender.
Add Dumplings to Soup: Bring the soup to a gentle simmer. Use a small spoon to drop tablespoon-sized amounts of dumpling dough over the surface of the simmering soup. The dumplings will double in size as they cook.
Cover and Cook: Cover the pot with a lid and let the dumplings simmer on low heat for about 12 to 15 minutes. Check for doneness by gently stirring to break them apart.
Finish and Serve: Once the dumplings are tender, chop the cooled chicken and add it back to the pot. Taste and adjust the seasonings if needed. Serve the Chicken and Dumplings warm, garnished with fresh chopped parsley on top.