Select your baking dishes. You can either use ten 12-ounce ramekins or gratin dishes, or six 12-ounce ramekins and one 1 1/2-quart gratin dish.
Preheat oven to 400 degrees.
Thaw puff pastry according to package directions.
Heat 1 1/2 tablespoons olive oil in a 4-quart or larger Dutch oven over medium-high heat. Season chicken with 1 teaspoon salt, 3/4 teaspoon pepper, and cayenne. Place half the chicken in Dutch oven and cook until browned on both sides, about 3 to 4 minutes. Remove chicken and set aside; repeat with remaining olive oil and chicken. Add butter to pan and, when melted, add mushrooms. Cook until mushrooms are well caramelized, 3 to 4 minutes, then add onions, carrots, celery, and garlic and cook, stirring occasionally until lightly caramelized, 4 to 5 minutes. Add remaining 2 teaspoons salt and 3/4 teaspoon pepper. Add flour and cook, stirring, 1 minute. Add stock, 2 tablespoons cream, potatoes, thyme, and browned chicken. Bring to a boil, reduce heat just to a simmer, cover, and cook, stirring occasionally, until chicken and potatoes are tender, about 25 minutes. Gently stir in peas; remove pan from heat.
On a lightly floured work surface, roll out one puff pastry sheet into a 12-inch square. Set ramekin or gratin dish upside down on pastry and, using a sharp knife and the shape of the dish as your guide, cut pastry about 1/2 inch larger than the dish. Set the dish on a rimmed baking sheet and ladle in filling. Top each dish with pastry, pressing along edges to seal. Repeat with remaining pastry sheet and filling. At this point, you can cover and refrigerate potpies until ready to bake, up to one week, or freeze for up to 2 months.
Whisk egg yolks with 1/2 cup cream and brush over the tops of puff pastry. Bake until pastry is puffed and golden brown, about 20 minutes. Allow to cool briefly before serving.
Cook's Note: You can fill whatever size ovenproof containers you have for this dish. It can be served in individual portions, or you can make larger ones to be served family-style. Simply roll your puff pastry to fit your dish.