Cut the chicken into ¾–1-inch cubes. Season with salt and black pepper.
Heat the olive oil in a large pot (we use a 6-quart enameled cast iron Dutch oven) over medium heat. When the oil is hot but not smoking, add the chicken in a single layer. Cook for 3–4 minutes, stirring occasionally, until browned on all sides.
Remove the pot from the heat. Using a slotted spoon, transfer the chicken to a plate. Loosely tent with foil and set aside. Don’t worry if it isn’t fully cooked; it will finish cooking in the soup.
Do not wash or wipe the pot. Return it to medium heat and add the butter. When melted, add the onions, carrots, and celery. Cook for 3–4 minutes, stirring often, until the onions are translucent.
Stir in the garlic and thyme. Cook for 1 minute, stirring often.
Slowly pour in the chicken broth while stirring and gently scraping any bits from the bottom of the pot.
Return the chicken to the pot along with any juices collected on the plate. Raise the heat and bring the broth to a boil, then lower to medium-low. Cover and simmer for 10 minutes, stirring occasionally.
While the soup simmers, cut the biscuits into quarters.
Gently drop the biscuit pieces into the simmering soup in a single layer. If the liquid stops simmering, raise the heat slightly to maintain a gentle simmer.
Press the biscuit pieces down gently with the back of a wooden spoon to submerge them. Lower the heat to medium-low, cover, and cook for 15–17 minutes, or until the dumplings are cooked through.
Check occasionally to make sure the broth is simmering (not boiling vigorously) and gently stir to prevent sticking. Avoid lifting the lid too often; the trapped steam helps the dumplings cook evenly.
Once the dumplings are done, remove the pot from the heat. When the soup stops simmering, stir in the half & half. Cover and warm gently over low heat for about 5 minutes. Do not let it boil.
Taste and add salt if needed. As a reference, we did not add extra to ours. Serve hot and garnish with chopped fresh parsley, if desired.