Season chicken with salt and pepper. Heat oil in Dutch oven over medium-high heat. Sear chicken, skin-side down, to deep brown, 8 to 10 minutes. Flip and cook another 5 to 8 minutes. Transfer to a plate, and pour off all fat into a measuring cup

Leaving browned bits in the pot, return 2 tablespoons of fat to the pot. Add onions, celery and half the carrots. Season with salt and pepper and cook on medium heat, stirring to scrape up all the bits on the bottom of the pot. Cook until vegetables start to soften. Return chicken to the pot along with thyme and 8 cups of water. Simmer, uncovered, until chicken is completely tender and liquid has reduced by about 1/4, about 30 to 40 minutes. Transfer chicken to a plate. Strain the stock.

Heat remaining 3 tablespoons chicken fat with 1 tablespoon butter over medium heat. Sprinkle in flour and stir constantly until pale golden brown. Slowly whisk in reserved chicken stock until no lumps remain and bring to a boil. Add leeks and remaining carrots, season with salt and pepper and lower the heat to simmer.

Remove and discard the skin and bones. Shred meat and add to pot. Cook, stirring occasionally, until liquid is thickened and carrots and leeks are tender, 10 to 12 minutes.

Dumplings: Whisk flour, baking powder, salt and pepper. Whisk buttermilk and egg. Add to dry ingredients, then melted butter. Stir just to combine.

Drop quarter-size dollops into the pot spacing apart. Cover and reduce heat to low. Cook until dumplings are puffed and cooked through, 18 to 22 minutes.