Line a baking sheet with parchment paper. On a lightly floured surface, roll out the pastry dough with a floured rolling pin. Transfer to the lined baking sheet, cover with plastic wrap and refrigerate while preparing the pie filling.
Combine the broth, peppercorns, thyme sprigs, bay leaves and quartered onion in a large pot over high heat. Bring to a boil, reduce the heat to a simmer and simmer for 30 minutes. Add the chicken breasts. Raise the heat and return to a boil. Reduce the heat and simmer until the chicken is just cooked through, 10 to 15 minutes.
Transfer the chicken to a plate to cool. Strain the stock and set aside 2 1/2 cups; reserve the remaining stock for another use. When the chicken is cool to the touch, cut it into bite-size chunks and set aside.
Preheat the oven to 375 degrees F.
Melt the butter in a large skillet over medium heat. Add the pearl onions and carrots and cook, stirring occasionally, for about 5 minutes. Add the mushrooms and peas and cook another 5 minutes. Stir in the 5 tablespoons flour and cook 1 minute more. Add the reserved stock and the milk, stirring constantly. Cook until the sauce begins to thicken, about 5 minutes. Remove from the heat and add the chicken, chopped thyme, salt and pepper.
Remove the pastry from the refrigerator and cut into 4 circles using one of the 2-cup ovenproof bowls as a guide.
Spoon the filling into four 2-cup ovenproof bowls, then place 1 dough circle on each pot pie, pressing around the edges to seal the pastry to the bowl. Cut a vent in the center of each. Using a pastry brush, brush the outer surface of each pie with egg wash. Bake until the pastry is puffed and dark golden brown, about 35 minutes.