In a 4-qt. saucepan, cook potatoes in generously salted water until tender, about 20 minutes. Drain and pass in a potato ricer back into the pan. Add 1 cup milk, 6 tablespoons of butter, salt, and pepper and stir to combine. Keep warm.
Melt 2 tablespoons butter in an 8-qt. saucepan over medium-high. Season chicken with salt and cook, stirring occasionally, 5 to 6 minutes. Using a slotted spoon, transfer chicken to a bowl.
Melt remaining 6 tablespoons butter in saucepan over medium-high. Add mushrooms, onion, and pepper and cook, stirring, until soft, 3 to 4 minutes. Add flour and cook 2 minutes more. Add remaining 3 1⁄2 cups milk and cook, stirring, until thick, 6 to 8 minutes. Stir in reserved chicken and its juices, along with the peas, and cook 2 minutes longer. Season with salt and pepper and divide among 8 8-oz. ramekins or spread evenly into a 9-by-13-inch baking dish. Top evenly with mashed potatoes and set aside.
Heat oven to 375°. Using a rolling pin, roll puff pastry until 1⁄8" thick. Using a 4-inch round cutter, cut out eight circles (if baking in a 9-by-13-inch baking dish, cut pastry into a 10-by-14-inch rectangle). Using a pastry brush, brush the rim of each ramekin with egg wash and cover with a pastry circle and press lightly around the edges with a fork to adhere (if using a baking dish, follow the same procedure). Brush pastry with egg wash and place on a baking sheet. Bake until golden brown and filling is warmed through, about 35 minutes. Let cool 10 minutes before serving.