1. Melt the butter with the olive oil in a large heavy pot or Dutch oven over medium heat.

2. Add the onion, carrots, and celery and cook, stirring often, until softened and lightly golden, about 6 to 8 minutes.

3. Stir in the garlic, salt, pepper, thyme, and dried parsley and cook until fragrant, about 1 minute.

4. Sprinkle the flour over the vegetables and stir until everything is evenly coated and the flour looks slightly toasted, about 2 minutes.

5. Slowly whisk in the chicken broth, a little at a time, stirring constantly to avoid lumps, then add the milk and bay leaf.

6. Bring the mixture to a gentle simmer and cook, stirring occasionally, until slightly thickened and velvety, 8 to 10 minutes.

7. Stir in the shredded chicken and frozen peas, return to a gentle simmer, and reduce the heat to low so it bubbles softly.

8. In a medium bowl whisk together the all-purpose flour for dumplings, baking powder, and kosher salt for dumplings.

9. Cut in the cold butter with a pastry cutter or your fingertips until the mixture looks like coarse crumbs.

10. Stir in the buttermilk, egg, and chopped parsley just until a thick, sticky dough forms without dry patches.

11. Use a medium cookie scoop or two spoons to drop heaping tablespoonfuls of dough over the surface of the simmering chicken mixture, leaving a little space between each dumpling.

12. Cover the pot tightly with a lid and cook on low, without lifting the lid, until the dumplings are puffed and cooked through, 15 to 18 minutes.

13. Remove the lid, discard the bay leaf, and gently stir around the edges to thicken the broth if needed while keeping the dumplings mostly on top.

14. Taste and adjust seasoning with more salt and pepper if you like, then ladle the chicken and dumplings into bowls and garnish with extra parsley.