In an 8-quart or larger pot, combine the chicken, carrot pieces, onion pieces, celery pieces, garlic, thyme, rosemary, bay leaves, stock, water, 2 teaspoons salt, some black pepper and crushed red pepper. Bring to a simmer and cook until the chicken is tender, about 1 hour.
While the chicken is cooking, make the dumplings: In a medium mixing bowl, mix the flour with remaining 1 1/2 teaspoons salt, baking powder and chopped herbs well with a fork. Add shortening and work it into the flour with your fingers, a pastry blender or a fork until the mixture resembles coarse crumbs. Add the milk and stir with a fork until incorporated.
Gather the dough and press gently into a ball. Turn the dough out onto a lightly floured surface and pat together. Knead the dough gently 6 or 7 times then divide it into 3 pieces. Working with 1 piece at a time, roll the dough out with a lightly floured rolling pin to a 1/16-inch thickness. Cut the dough into 1 1/2-inch wide strips then cut the strips crosswise into roughly 1 1/2-inch squares. Transfer the dumplings to a parchment-lined baking sheet and repeat with the remaining dough, putting a piece of parchment paper between the layers. Refrigerate the dumplings until ready to use, up to 1 day in advance.
Remove the chicken from the pot and set aside until cool enough to handle. Discard the skin, remove the meat from the bones, and reserve the meat. Discard the bones. Strain the broth into another large pot or container and reserve. Discard the solids.
Return the original pot to the stove over medium heat. Add the butter and, when melted, add the chopped carrots, onions and celery. Cook for 2 minutes, then add the strained broth and increase the heat to high. When the broth comes to a boil, add the dumplings in batches, stirring between additions. When the liquid returns to a boil, reduce heat to low, cover pot and simmer for 15 minutes. Stir the dumplings once, very gently, to prevent them from sticking to the bottom of the pan and continue to cook, covered and undisturbed, until the dumplings are very tender, about 15 minutes longer.
Stir in the reserved chicken and the peas. Return the sauce to a gentle boil, cover and remove from heat. Let the chicken and dumplings sit, covered, for 20 minutes before serving.
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