Heat the butter and the oil together in a large, lidded saucepan or Dutch oven over a medium high heat. Once the butter has melted, stir in the carrots, onion, celery and salt. Gently fry for 8-10 minutes until soft, and just starting to color.
Stir in the garlic, and cook for a further 2 minutes until aromatic before adding the chicken, the thyme and the bay leaves.
Add the flour, stirring so that the flour coats everything in the pot. Cook for a further 2 minutes.
Gradually pour the chicken broth into the pan, scraping any browned bits off the bottom of the pot. Stir in the heavy cream or half and half, and reduce the heat to a simmer, leaving to cook uncovered for 15 minutes.
Meanwhile, in a mixing bowl whisk together the flour, baking powder and salt with a fork. Add the heavy cream and melted butter, and bring together into a sticky yet uniform dough.
Remove the chicken thighs from the soup and shred on a cutting board with two forks. Return to the soup and stir.
Spoon the dumpling mixture into the soup, allowing for 2 dumplings per person. Cover, and continue to simmer for 10 minutes.
Add the frozen peas, and stir gently to turn over the dumplings so they cook through. Cook for a further 5 minutes.
Season to taste with black pepper and more soup if needed before serving.