Combine the whole chicken, water, bay leaves, whole black peppercorns, and 1.5 teaspoons kosher salt in a large pot. Bring just to a simmer over medium-high heat, then reduce to low, skim foam, and cook gently until the chicken is tender and the thighs read 175°F, 45–60 minutes.
Transfer the chicken to a tray to cool, 10 minutes. Strain the broth into a bowl or measuring jug; you should have about 8 cups—add water to reach 8 cups if needed.
When cool enough to handle, discard the skin and bones and shred the chicken into bite-size pieces, 5–10 minutes.
In a Dutch oven, melt 2 tablespoons unsalted butter over medium heat until foamy, 30–60 seconds. Add the onion, carrots, and celery; cook, stirring, until softened and translucent, 6–8 minutes.
Sprinkle 0.25 cup all-purpose flour over the vegetables and cook, stirring constantly, until the flour turns pale blond and smells nutty, about 2 minutes.
Gradually whisk in the 8 cups reserved broth until smooth. Add the dried thyme, ground black pepper, and 0.5 teaspoon kosher salt. Bring to a gentle simmer and cook until lightly thickened, 5 minutes.
Stir in the shredded chicken and return to a bare simmer over medium-low heat, 5 minutes. Adjust heat to maintain gentle bubbling.
Make the dumpling batter: In a medium bowl, whisk together 2 cups all-purpose flour, baking powder, baking soda, and 1 teaspoon kosher salt. Add the buttermilk and 4 tablespoons melted, cooled unsalted butter; stir just until a thick, sticky batter forms. Let rest 5 minutes.
With the stew at a gentle simmer (not boiling hard), drop heaping tablespoonfuls of batter onto the surface, spacing them slightly; you should get 18–24 dumplings. Cover the pot tightly and cook over low heat without lifting the lid for 15 minutes.
Uncover and check doneness: a skewer inserted into a dumpling should come out clean and the centers should look set; if needed, simmer uncovered 3–5 minutes more to finish. Taste the broth and dumplings and add the remaining 0.5 teaspoon kosher salt if needed. Let rest off heat 5 minutes to settle.
Ladle into bowls and sprinkle with chopped fresh parsley (for serving). Serve hot.