1 - In a large saucepan over medium heat, bring chicken, 2 cups milk, 3 sprigs thyme, and 3/4 teaspoon each salt and pepper to a simmer. Continue to simmer until chicken is cooked, 15 to 20 minutes, flipping halfway through. Transfer chicken to a cutting board and cool. Dice meat and set aside. Strain hot milk from saucepan and set aside 1 cup. Discard the remainder of contents.
2 - Preheat oven to 400F. Meanwhile, in a large nonstick skillet over medium-high heat, heat oil. Saute mushrooms until soft, about 4 minutes. Add carrots and onion, reduce heat to medium, and cook until carrots are soft, about 10 minutes. Season with salt and pepper. Remove leaves from remaining thyme sprigs and add to skillet, along with peas and reserved chicken and milk; bring to a simmer.
3 - In a small dish, stir together cornstarch and 3 tablespoons cold water. Add to skillet and stir over high until mixture thickens, 2 to 3 minutes. Remove skillet from heat and set aside.
4 - In a small bowl, lightly beat egg white and remaining 1 tablespoon milk. Place empanada disks on 2 parchment-paper-lined baking pans. Divide chicken filling among empanada disks, leaving 1/2-inch borders around edges. Dampen edges with egg wash and fold dough over to create turnovers. Crimp edges with a fork to seal. Lightly brush tops of turnovers with egg wash. Bake until crusts are golden brown, about 15 minutes, turning pans halfway through.
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