In a medium bowl, season the chicken with salt and pepper. Toss with 1/2 cup of flour until all pieces are evenly coated. This light coating not only seasons the chicken but also helps thicken the broth later.

Heat 1 tablespoon of olive oil in a large pot or Dutch oven over medium-high heat. Add the floured chicken and cook until lightly browned on the outside, about 3-5 minutes, stirring occasionally. You don’t need to cook the chicken through at this stage—it will finish cooking in the broth. Remove the chicken to a plate and set aside.

Reduce the heat to medium and add the remaining olive oil along with the butter. Once melted, add the carrots, celery, and onion. Sauté until softened, about 5 minutes, stirring occasionally. This step releases the natural sweetness of the vegetables and adds depth to the final dish.

Stir in the garlic and cook for 30 seconds until fragrant. Add the bay leaves, dried thyme, and chicken broth, scraping up any browned bits from the bottom of the pot. These bits are full of flavor and will enrich your broth.

Return the browned chicken to the pot. Bring the mixture to a simmer and let it cook gently until the chicken is tender and the flavors have melded, about 10 minutes. Stir in the heavy cream and thawed peas. Taste and adjust seasoning with salt and pepper as needed.

While the soup is simmering, prepare the dumplings. In a medium bowl, whisk together the flour, baking powder, and salt. In a separate small bowl, beat the eggs and buttermilk until combined. Pour the wet ingredients into the dry mixture and stir just until no streaks of flour remain. Avoid overmixing—this helps keep the dumplings light and tender.

Bring the soup to a gentle simmer. Using a spoon, drop the dumpling dough by tablespoons into the broth, giving them space to puff up. Cover the pot with a lid, reduce the heat to low, and let simmer for 15 minutes. Resist the temptation to lift the lid too often, as the steam is essential for cooking the dumplings properly.

Once the dumplings are puffy and cooked through, remove the bay leaves. Taste the broth one last time for seasoning. Serve hot in deep bowls, making sure each serving has a generous portion of chicken, vegetables, dumplings, and creamy broth.