01 - In a large bowl, combine the all-purpose flour and salt. Incorporate the cold, cubed butter using a pastry blender or your fingertips until the mixture resembles coarse crumbs. In a separate small bowl, whisk together the egg, cold water, and white vinegar. Gradually add this liquid mixture to the flour mixture, stirring until a cohesive dough forms. Briefly knead the dough on a lightly floured surface, shape it into a disk, wrap it in plastic, and refrigerate for at least 30 minutes to firm up.
02 - Melt the unsalted butter in a skillet over medium heat. Add the finely diced onion, diced carrot, and diced celery. Sauté for approximately 5 minutes until the vegetables have softened. Stir in the minced garlic and cook for an additional minute until fragrant. Sprinkle the all-purpose flour over the vegetables and stir to coat, cooking for 2 minutes to eliminate the raw flour taste. Gradually whisk in the chicken broth and whole milk, stirring continuously until the mixture thickens into a sauce, which should take about 3–4 minutes. Fold in the shredded cooked chicken breast, frozen peas, dried thyme, black pepper, and salt. Remove the skillet from the heat and allow the filling to cool completely.
03 - Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper. On a lightly floured surface, roll out the chilled empanada dough to a uniform thickness of 1/8 inch. Use a round cutter (4 to 5 inches in diameter) to cut out circles from the dough. Spoon approximately 2 tablespoons of the cooled filling onto the center of each dough circle. Fold the dough over the filling to create a half-moon shape, and press the edges firmly to seal. Crimp the edges with a fork. Brush the tops of the assembled empanadas with the beaten egg wash.
04 - Place the prepared empanadas onto the lined baking sheet. Bake in the preheated oven for 18 to 22 minutes, or until the crust is beautifully golden brown and puffed. Allow the empanadas to cool slightly before serving to prevent burns.