Cook chicken and make stock by combining all ingredients in an 8 quart or larger heavy-bottomed pot. Bring to a simmer and cook until the chicken is falling off the bones tender, about 1 hour. Remove the chicken from the pot and set aside until it's cool enough to handle. While chicken is cooking, prepare the dumplings.
Make the dumplings: In a medium bowl, mix the flour, salt, baking powder, and the fresh herbs with a fork. Add the shortening and cut it into the flour mixture with a pastry blender or two knives (the mixture will look like coarse crumbs). Add the milk and stir until combined. Gather the dough into a ball. Flour your countertop. Turn your dough out on the floured counter. Knead the dough gently until it comes together about 6-7 times. Divide the dough into thirds. Roll out the dough, one third at a time, until it's 1/8 inch thick. Cut the rolled out dough into roughly 1-inch strips then cut the strips crosswise into roughly 1 1/2 inch rectangles. (Use a floured knife or pastry cutter to do this don't worry if your pieces aren't exactly the same size...they're handmade!). Place cut pieces on a lightly floured baking sheet. Repeat the rolling and cutting with the rest of the dough then stack the other layers of cut dough between sheets of parchment paper on the same baking sheet. You can refrigerate the dumplings at this point if you aren't serving right away, for up to one day.
Remove the skin and bones from the cooked chicken and shred or cut it into bite-size pieces; set aside.
Strain the broth into another large bowl or pot and discard the cooked veggies. Set aside. Add the butter to the original large pot and saute the diced carrot, celery and onion for 5-7 minutes on medium heat. Carefully add back the strained broth to the pot, increasing the heat to high, and bring to a boil. When the stock is boiling add the dumplings in batches, stirring between additions. Return the stock to a boil, then reduce the heat to low and cover and simmer for 15 minutes. Stir the dumplings. Cover and cook another 15 minutes.
Stir in the chicken and the peas. Bring the stock back to a gentle boil then cover the pot and remove it from heat. Allow the Chicken and Dumplings to sit, covered for 20 minutes before serving.