Heat 1 Tbsp of the butter and the oil in a large heavy-bottomed pot set over medium heat. Pat the pieces of chicken dry with a paper towel and season well with salt and pepper. Working in batches, brown the chicken, transferring the cooked pieces to a plate.
Once all of the chicken is browned, add the remaining tablespoon of butter to the pot. Add the onion, celery, carrots and thyme and stir well. Cover and cook for 5 minutes, stirring occasionally, until the vegetables begin to soften.
Add the flour and cook, stirring, for 1 minute. Add the stock and bring to a boil, stirring constantly. Nestle the chicken back in the pot, reduce the heat to medium-low, cover the pot and cook for 20 minutes, stirring occasionally.
Just before the chicken is done cooking, make the dumplings. Whisk together the flour, baking powder, dill weed and salt in a mixing bowl. Add the melted butter and milk to the dry ingredients and gently mix with a wooden spoon or spatula just until the mixture comes together. Do not overmix or your dumplings will be dense.
Stir the peas and lemon juice into the pot. Season to taste with salt and pepper. Drop small balls of dumpling batter (8–10 total) into the simmering liquid, keeping at least 1” between them. Cover the pot and simmer until the chicken is tender and the dumplings are puffed and firm, about 15-20 minutes.