Preheat the oven to 350 degrees F.
Melt the butter in a saucepan over medium heat, add the chopped onion and cook until tender, about 5 minutes. Stir in the carrots and celery and cook for 2 minutes. Add the flour, stirring to coat the vegetables, and cook for 2 minutes. Add the chicken broth and 1 teaspoon each kosher salt and pepper and bring to a simmer. Simmer until slightly thickened, about 10 minutes. Stir in the chicken, peas and parsley. Pour the mixture into a 9-inch deep-dish pie plate.
Put the baking soda, 2 teaspoons kosher salt and 8 cups water in a large, heavy pot and bring to a boil.
While the water comes to a boil, divide the pizza dough into 4 equal parts (each portion weighing about 6 ounces). Roll each portion of dough into a snake about 3 feet long. Pick up the ends of the dough, forming a U shape. Next, twist the ends of the dough together twice. Bring the twisted end of the dough towards you and fold it down onto the bottom curve. Press each of the ends into the dough, using a bit of water to help them stick. Repeat with the remaining dough pieces.
Use a slotted spoon or mesh strainer to carefully lower one of the shaped pretzels into the boiling water. Cook for 30 seconds, then gently flip and cook for another 30 seconds. Remove the boiled pretzel and drain on a cutting board. Repeat this process with the remaining 3 pretzels.
Arrange the boiled pretzels on top of the pot pie mixture, with the rounded edges touching the edge of the pie dish and the tail ends of the pretzels hanging over. Whisk together the egg and 1 tablespoon water in a small bowl. Brush each pretzel with the egg wash and sprinkle with the pretzel salt.
Bake until the pretzels are dark brown and the filling is bubbling and hot throughout, 45 to 50 minutes.
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