In large saucepan over medium heat, bring chick, 2 cups milk, 3 sprigs thyme, and 3/4 tsp s&p to simmer. Continue to simmer until chick cooked, 15-20 minutes, flipping halfway through. Transfer chicken to cutting board and cool. Dice meat and set aside. Strain hot milk from saucepan and set aside 1 cup. Discard remainder contents.
Preheat oven 400*F. In large nonstick skillent over medium high heat, heat oil. Saute mushromms until soft, about 4 minutes. Add carrots and onion, reduce heat to mdeium, and cook until carrots are soft, about 10 minutes. Seat with s&p. Remove leaves from remaining thyme and add to skillet, along with peas and reserved chicken and milk, bring to simmer.
In small dish, stir together cornstarch and 3 Tbsp cold water. Add to skillet and stir over high heat until mixture thickens, 2-3 minutes. Remove skillet from heat and set aside.
In small bowl, lightly beat egg white and remaining 1 Tbsp milk. Place empanada disks on 2 parchment lined baking pans. Divide chicken filling amon empanada disks, leaving 1/2-inch border around edges. Dampen edges with egg wash and fold dough over to create turnovers. Crimp edges with fork to seal. Light brush tops with egg wash. Bake until golden, about 15 minutes, turning pans halfway through.
N.B. Empanada disks are lower in fat/calories than pie dough.
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