Warm oil in a large pot over medium heat.
Sauté onion, carrots, and celery until softened, about 5 minutes.
Stir in garlic and cook until fragrant, 30 seconds.
Sprinkle in flour and cook, stirring, 1 minute.
Whisk in chicken broth gradually, scraping the pot. Bring to a gentle simmer.
Add chicken, thyme, salt, and pepper. Simmer 8 to 10 minutes.
Mix dumplings: combine baking mix, milk, and cheddar until just moistened.
Drop heaping tablespoons of dough onto the simmering soup surface.
Cover and cook 12 to 14 minutes without lifting the lid.
Uncover, stir in half-and-half, and simmer 2 minutes more.
Finish with parsley, taste, and adjust seasoning before serving.