Heat olive oil in a large pot (at least 6-quart) over medium heat. Add the celery, carrots, and onion, and cook for about 3 minutes until slightly softened. Add the garlic and sauté for 30 seconds. Stir in the butter and flour, whisking for 1 minute to form a light golden roux.

Gradually whisk in the chicken broth and bouillon, scraping up any bits from the bottom. Add thyme, sage, rosemary, salt, and pepper. Stir well.

Add the whole chicken pieces to the pot and bring the mixture to a boil. Cover, reduce heat to low, and simmer for about 10 minutes, or until the chicken reaches 165°F internally. Remove the chicken to a cutting board and let it rest.

Stir in the evaporated milk and peas. Let simmer gently (small bubbles, around 185-200°F) while you prepare the dumplings.

In a mixing bowl, combine flour, baking powder, and salt. Make a well in the center, then add milk, sour cream, and melted butter. Stir until just combined into a thick, slightly sticky batter - do not overmix.

Bring the soup to a gentle simmer. Drop tablespoon-sized portions of the dumpling dough onto the surface of the soup, spacing them about 1 inch apart. Cover and simmer on low for 12 to 15 minutes without removing the lid.

Check one dumpling for doneness by cutting it in half - it should be fluffy throughout with no raw dough in the center. Shred the chicken and return it to the pot. Stir gently to combine. Garnish with fresh parsley and serve warm.