1. Cook the leeks: Melt butter in a medium skillet over medium heat. Add sliced leeks and kosher salt. Cook for 8–10 minutes, stirring frequently, until softened but not browned.

2. Prepare the sauce: Stir in the flour and cook for 1 minute. Gradually add chicken stock while stirring. Simmer for about 2 minutes until slightly thickened.

3. Make the filling: Remove from heat and stir in shredded chicken, sour cream, Dijon mustard, tarragon, thyme, and black pepper. Adjust seasoning to taste.

4. Cool the filling: Let the chicken and leek mixture cool for approximately 30 minutes, until completely cold or at least room temperature.

5. Preheat the oven: Preheat to 400°F (200°C). Line two baking sheets with parchment paper.

6. Prepare the pastry: On a lightly floured surface, roll each puff pastry sheet to approximately 13 × 13 inches. Cut 9–10 bottom circles using a 3½-inch cutter and matching top circles using a 4-inch cutter.

7. Fill the hand pies: Place about 2 tablespoons of cooled chicken filling in the center of each smaller pastry circle, leaving a border around the edges.

8. Seal the pastry: Beat the egg with 1 teaspoon water. Brush the pastry borders with egg wash, cover with the larger circles, and press the edges firmly. Crimp with a fork to seal.

9. Add the finishing touches: Brush the tops with egg wash and make small steam vents using a sharp knife. Garnish with fresh thyme sprigs if desired.

10. Bake until golden: Bake for approximately 22 minutes, rotating the pans after 15 minutes, until the puff pastry is puffed, crispy, and deep golden brown.

11. Cool and serve: Let the hand pies rest for 5 minutes before serving. Enjoy warm!