Line a baking sheet with parchment paper. On a lightly floured surface, roll out the dough with a floured rolling pin into a 12-inch square. Transfer to the prepared baking sheet, cover with plastic wrap and refrigerate while you make the filling.
Cook the chicken.
Combine the chicken broth, peppercorns, thyme sprigs, bay leaves and quartered onion in a large pot over high heat. Bring to a boil, then reduce the heat and simmer 30 minutes.
Add the chicken to the pot. Increase the heat and return to a boil, then reduce the heat and simmer until the chicken is just cooked through, 10 to 15 minutes. Using tongs, transfer the chicken to a plate to cool. Strain the stock through a fine-mesh sieve. You¿ll need 21/2 cups for the filling; reserve the rest for another use. When the chicken is cool to the touch, cut it into bite-size chunks and set aside.
Make the filling.
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, until softened, about 5 minutes.
Add the mushrooms and peas and cook until softened, another 5 minutes. Stir in 5 tablespoons flour and cook, stirring, 1 more minute. Add 2 1/2 cups chicken stock and the milk, stirring constantly. Cook until thickened, about 5 minutes. Remove the skillet from the heat. Add the chicken and chopped thyme; season with salt and pepper.
Form the pot pies.
Set out four 2-cup ovenproof bowls or ramekins. Transfer the pastry to a floured surface. Using a paring knife, cut out 4 rounds of dough that are slightly larger than the mouth of the bowls.
Place the bowls on a baking sheet and spoon in the filling. Place 1 dough round on each pot pie, pressing the pastry against the side of the bowl to seal. Brush the dough with the beaten egg using a pastry brush. Cut a slit or hole in the center of the dough to vent. Bake until the pastry is puffed and dark golden brown, about 35 minutes.