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Step 1
Preheat the oven to 400 degrees F (200 degrees C). Line a large baking sheet with foil. Lightly grease six 12-ounce oven-safe ramekins and place on the baking sheet for easy cleanup.
Step 2
Melt 4 tablespoons butter in a pot over medium-low heat. Add leeks, onion, 1/3 of the chopped celery, and garlic; saute until tender, about 5 minutes. Add broth, potatoes, and bay leaves. Season with pepper, salt, thyme, sage, and rosemary; bring to a boil. Reduce heat, cover, and simmer for at least 25 minutes.
Step 3
Remove bay leaves. Use an immersion blender to emulsify the liquid.
Step 4
Slowly stir in heavy cream. Add remaining 3 tablespoons butter with carrots, peas, and remaining celery. Stir in chicken. Cover and simmer for about 15 minutes.
Step 5
Mix egg and water together and set aside for egg wash.
Step 6
Roll puff pastry out slightly. Flip one of the prepared ramekins upside-down on the pastry; cut out circles 1/2 inch larger than the ramekin.
Step 7
Fill each ramekin 3/4 full with the chicken mixture; do not fill to the top or it will make a mess in your oven. Place cut rounds of puff pastry over the tops and press around the edges to seal. Brush egg wash over each. Be sure to put a few knife slits on top of each pie to allow to vent.
Step 8
Bake in the preheated oven until the pastry is a golden brown, about 30 minutes. Let cool at room temperature for 10 to 12 minutes before serving.