Heat olive oil and butter in a large pot or Dutch oven over medium heat.
Add chicken, season lightly with salt and pepper, and cook 4–5 minutes until lightly browned.
Add onion, carrots, and celery. Cook 4–5 minutes until vegetables soften.
Stir in garlic, poultry seasoning, and thyme. Cook 30 seconds until fragrant.
Sprinkle flour over the mixture and stir well. Cook 1 minute to remove raw flour taste.
Slowly pour in chicken broth, stirring constantly until smooth.
Bring to a gentle boil, then reduce heat to a simmer.
Stir in milk or half-and-half and simmer 8–10 minutes until slightly thickened.
Evenly drop biscuit dough pieces over the surface of the soup.
Cover the pot and simmer 12–15 minutes without lifting the lid, until dumplings are puffed and cooked through.
Turn off heat and let rest 5 minutes. Stir gently, taste, and adjust seasoning before serving.