For the chicken: Add the chicken, carrots, celery, onions, thyme sprigs, bay leaf, 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 the butter to the remaining broth and let melt.
For the pastry: Whisk together the flour, 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, butter, salt and cayenne to a large bowl and beat with a hand mixer until mixed together. Add the 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 degrees F. Add one layer of dough strips to the bottom of a 4-quart rectangular casserole dish. Add a layer of peas, 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 butter squares and sprinkle with salt and pepper.
Bake for 1 hour and 45 minutes. Cool for 15 minutes before serving.