Prep the vegetables for the soup: dice the onion, carrots, and celery, mince the garlic.
Add butter and olive oil to a large pot or Dutch oven set over medium heat. Once heated and the butter has melted, add the onion, carrots, and celery and cook for 8 minutes. Add the garlic and cook for 1 minute more.
Scatter the dried spices over the vegetables: rubbed sage, marjoram, rosemary, thyme, white pepper, smoked paprika, nutmeg, and a big pinch of kosher salt.
Sprinkle in the flour and stir to combine everything. Cook for 2-3 minutes, stirring while the mixture thickens up.
Pour in roughly ½- to ⅔-cup of the broth to deglaze the pan, scraping up any stuck-on bits from the bottom. Add in the rest of the broth, along with the milk and peas. Adjust the heat so that everything simmers, uncovered, for at least 9-10 minutes. The longer you simmer during this phase the thicker the soup will become.
While the soup is simmering, shred the chicken and set aside, then prepare the dumplings. Whisk together Bisquick, milk, grated Parmesan, and garlic powder in a mixing bowl until evenly combined.
Once the soup has simmered, add the shredded chicken and parsley into the soup and stir. Scoop roughly 1 ½ Tablespoons of the dumpling mix and place them around the surface of the soup. Place the lid fully-on the pan, then cook for 15 minutes, aiming for it to simmer the whole time without coming to a big boil. Remove the lid and cook for an additional 5 minutes, or until the dumplings are cooked through (a toothpick inserted in the center of one comes out clean). Serve.