Bring chicken, leek, whole carrot, whole celery stalk, half of onion, garlic, thyme, parsley stems, bay leaf, peppercorns, and 3 quarts water to a simmer in a large wide pot. Cook, uncovered, until chicken is cooked through, 20–30 minutes. Transfer chicken to a plate and let sit until cool enough to handle. Shred meat, discarding anu excess fat. Cover meat with foil to keep warm; set aside.
Strain stock through a fine-mesh sieve into a large bowl; discard solids. Wipe out pot, pour stock back in, and bring to a simmer. Stir in 1 Tbsp. plus 1 tsp. salt.
Heat ¼ cup plus 1 Tbsp. butter in a large skillet over medium. Cook chopped carrot, chopped celery, and remaining onion, stirring occasionally, until vegetables are softened but not completely tender, 8–10 minutes; season with salt and pepper. Add ½ cup flour and cook, stirring, until vegetables are coated, about 1 minute. Scrape vegetable mixture into simmering stock and whisk until liquid is thickened and no lumps remain, then continue to cook until vegetables are tender, 10–15 minutes.
Meanwhile, whisk baking powder, baking soda, 1 tsp. salt, ¼ tsp. pepper, and remaining 1 cup flour in a medium bowl. Whisk buttermilk and remaining 3 Tbsp. butter in a small bowl. Fold buttermilk mixture into dry ingredients just until dough comes together (be careful not to overmix or dumplings will be tough).
Add cream and reserved chicken to stew and return to a simmer. Drop tablespoon-sized portions of dough into stew (don’t worry if they aren’t perfect, they will puff up as they cook). Reduce heat to low, cover, and cook 10 minutes. Uncover and check dumplings for doneness: They should be about 5 times larger and should cover entire surface of stew. To be extra sure, you can remove a dumpling and cut it in half to see if it’s cooked through; the interior should look like a soft dinner roll. If they’re not fully cooked, cover and cook 2 minutes more.
Divide chicken soup and dumplings among bowls, then top with chives.