Place chicken in a large stockpot and add enough water to cover by 2 inches. Bring to a boil over medium-high heat, reduce to a simmer, and cook until chicken is cooked through (until an instant-read thermometer inserted into the thickest part registers 165 degrees), about 50 minutes to 1 hour. Use tongs to remove chicken. When cool enough to handle, pull off and discard the skin, then separate the meat from the bones and tear into bite-size pieces. Reserve 2 cups of stock for the pot pie and the rest for another use.
In a large heavy-bottomed pot, melt butter over medium-high heat. Add potatoes, carrots, and onion. Cook, stirring occasionally, until potatoes and carrots are fork-tender, 10 to 12 minutes. Add mushrooms, cooking until mushrooms are heated through. Add cognac and cook for 2 minutes. Add flour and cook, stirring, for 1 minute. Pour in reserved 2 cups chicken stock and milk. Bring to a simmer; cook, stirring constantly, until thick and bubbly, 2 to 3 minutes. Stir in chicken, peas, parsley, and thyme. Season with salt and pepper. Ladle into six (12-ounce) ramekins, let cool.
Roll out thawed puff pastry on a lightly floured surface to a 1/4-inch thickness. Cut out six circles that are 1 inch wider than ramekins or bowls being used. Cut a 1/2-inch vent hole in the center of each circle. Transfer puff pastry circles to a parchment-lined baking sheet and freeze for 20 minutes.
In a small bowl, whisk together egg, egg yolk, and 1 teaspoon water. Remove puff pastry circles from freezer. Top each ramekin with a puff pastry circle, crimping edges over sides to seal, and brush with egg wash. Transfer to refrigerator and chill for 30 minutes.
Preheat oven to 425 degrees. Transfer ramekins to a rimmed baking sheet. Bake until puff pastry is deep golden brown and filling is bubbling, 20 to 25 minutes.