For the chicken: Add the chicken, carrots, celery, onions, thyme sprigs, bay leaf and 1 tablepoon each salt and pepper to a large pot. Add water to cover and bring to a boil. Reduce the heat to low and simmer gently until the chicken is cooked through, about 45 minutes.
Remove the chicken from the pot to a baking sheet and set aside until cool. Shred the meat; discard the skin and bones. Remove the vegetables from the pot and discard. Stir in 1 stick of butter to the remaining broth and let melt.
For the pastry: Whisk together 3 cups flour, one teaspoon salt and baking powder. Cut in the shortening with a pastry blender until the size of coarse cornmeal. Sprinkle in 1 to 2 tablespoons of ice water and toss with a fork; repeat until the dough is moistened. Add some flour to your work surface and knead the dough gently with your hands to make into a ball of dough. Form into 2 balls and roll out with a floured rolling pin until 1/8-inch thick. Cut into long 1/2-inch-wide strips.
For the cheese straw: Add the cheese, 1 stick butter, 1 teaspoon salt and cayenne to a large bowl and beat with a hand mixer until mixed together. Add 1 1/2 cup flour and continue with the hand mixer until a smooth dough is formed. Roll out on floured work surface until 1/4-inch thick, and then cut into long 1/2-inch-wide strips.
To layer the pot pie: Preheat the oven to 350°F. Add one layer of dough strips to the bottom of a 4-quart rectangular casserole dish. Add a layer of peas, cooked carrots and chicken, and then another layer of pastry strips, peas, carrots and chicken. Repeat until you reach the top of the casserole dish. Top with a lattice crust of cheese straws. Pour in the warm broth to just below the top of the casserole dish (you will have leftover broth). Top with the 1/4 stick butter squares and sprinkle with salt and pepper.
Bake for 1 hour and 15 minutes. Cool for 15 minutes before serving.