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Preheat the oven to 425 degrees F. Unroll the pie dough onto a baking sheet and sprinkle with pepper and a pinch of poultry seasoning; cut into quarters. Bake until puffed and golden, about 10 minutes.
Meanwhile, melt the butter in a large pot over medium-high heat. Add the chicken and 1/2 teaspoon salt and cook, undisturbed, 2 minutes, then cook, stirring, 1 more minute. Transfer to a bowl.
Add the celery, onion, flour, 1/2 teaspoon salt and the remaining 1/2 teaspoon poultry seasoning to the pot and cook, stirring, 1 minute. Stir in 2 cups water, the broth, half-and-half and potatoes; cover and bring to a simmer. Reduce the heat to medium and simmer, partially covered, 10 minutes. Add the peas and carrots and simmer until the vegetables are tender, about 6 minutes. Return the chicken to the pot and simmer until cooked through, about 1 minute. Divide among bowls and top with the crust.
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