FOR THE STEW: Dry chicken with paper towels and season with salt and pepper. Heat 1 tablespoon oil in 12-inch skillet over medium-high heat until just smoking. Brown half of chicken lightly on both sides, 5 to 8 minutes; transfer to bowl. Repeat with 1 tablespoon more oil and remaining chicken; transfer to bowl.
Heat remaining tablespoon oil over medium-high heat until shimmering. Add onions, celery, garlic, tomato paste, and thyme, and cook until vegetables are softened and lightly browned, 8 to 10 minutes. Stir in flour and cook for 1 minutes. Slowly whisk in sherry, scraping up any browned bits. Whisk in 1 cup broth smoothing out any lumps; transfer to slow cooker.
Stir remaining 3-1/2 cups broth, carrots and bay leaves into slow cooker. Nestle browned chicken with any accumulated juice into slow cooker. Cover and cook 4 to 6 hours on low.
Transfer chicken to cutting board, let cool slightly, then shred into bite-size pieces. Let stew settle for 5 minutes, then remove fat from surface using large spoon. Discard bay leaves.
Stir shredded chicken, peas, and parsley into stew, and season with salt and pepper to taste. Cover and cook on high until simmering.
FOR THE DUMPLINGS: Whisk flour, baking powder, and salt together in large bowl. Microwave milk and butter together until warm, about 1 minute, then whisk to melt butter. Stir milk mixture into flour mixture until just incorporated and smooth.
Drop golf ball-size dumpling on top of simmering stew, leaving about 1/4-inch between each dumpling (you should have about 18). Cover and cook until dumplings have doubled in size, 25 to 35 minutes. Serve.