Add the olive oil and butter to a large soup pot and place it over medium-high heat.
Add the chicken and onion. Stir until the chicken is well coated.
Add the salt, parsley, thyme, and pepper. Cook, stirring occasionally, until the chicken is no longer pink and the onion is soft, about five minutes.
Add the garlic and flour. Cook, stirring often, for 1 minute.
Add one cup of the broth. Stir until smooth. Add the remaining broth, milk, and the peas and carrots. Stir well. Bring the liquid to a simmer.
As the soup is coming to a simmer, make the dumplings. Add the cheddar bay biscuit mix to a large mixing bowl. Add the water, cheese, and the cheddar bay flavor packet. Stir to combine.
Drop about 1 1/2 tablespoons worth of batter into the simmering liquid for each dumpling (I used a #40 scoop). Try to drop them so the dumplings are not touching. Use all the batter.
Cover the pot and cook for 12-15 minutes. Do NOT stir. And, do not take the lid off, so the dumplings can steam.
Break apart any dumplings that have stuck together. Gently stir to coat the dumplings in the soup. Test a dumpling by cutting it in half. If it is still doughy in the center, let the soup simmer another 2-3 minutes.