Combine chicken, 2 teaspoons salt, and 4 cups water in a medium saucepan. Bring to a boil, then reduce heat to medium-low and simmer, partially covered, until chicken is cooked, about 20 minutes. Transfer chicken to a plate; let cool 10 minutes. Discard skin and bones; shred meat into bite-size pieces (you should have 3 cups). Strain broth (you should have 4 cups; if not, add more water).

Melt 4 tablespoons butter in a large pot over medium-high heat. Add onion, carrots, celery, potato, and thyme. Cook, stirring, until onion is translucent, 3 to 4 minutes. Sprinkle with 1/4 cup flour; cook 30 seconds. Gradually stir in broth. Bring to a boil, then reduce heat to medium and simmer 2 minutes. Stir in chicken, peas, and 3 tablespoons parsley. Season with salt and pepper.

Meanwhile, whisk together remaining 1 2/3 cups flour, 3 tablespoons parsley, baking powder, 1 1/2 teaspoons salt, and 3/4 teaspoon pepper. Cut in remaining 4 tablespoons butter. Stir in milk until dough holds together. Drop 8 heaping spoonfuls of dough on top of chicken mixture. Cover and simmer until dumplings are cooked through, about 12 minutes. Serve with more chopped parsley and lemon wedges.