Rinse the chicken. Place it in a large saucepan or small pot, just large enough to hold the chicken comfortably, and add enough water to cover. Bring to a simmer over high heat. Reduce the heat and skim the foam that is floating on the top of the water. Add up to 1 tablespoon salt and 1/2 teaspoon pepper. Simmer until the chicken is cooked through, 40 to 45 minutes. Remove the chicken from the pot with a large slotted spoon and let cool. Pull the meat from the bones and shred it. Discard the skin and bones.
Strain the stock through a fine strainer. Measure 2 1/2 cups and set aside. Refrigerate or freeze the remaining stock for another use.
In a large saucepan over medium-low heat, melt the butter. Add the red onion, mushrooms, carrots, and parsnip, season with salt and pepper, and cook until softened, 5 to 7 minutes. Add the brandy and simmer until evaporated. Add the flour, stir to blend with the butter, and cook for 30 seconds. Add the 2 1/2 cups reserved chicken stock, bring to a simmer, and cook for 2 to 3 minutes. Add the milk, bring to a simmer, and cook until the vegetables are tender and the sauce has reduced to the consistency of a thick white sauce, 7 to 12 minutes. Add the peas, pearl onions, thyme or other herb, if using, parsley, nutmeg, and shredded chicken. Season with salt and pepper.
Preheat the oven to 375°F (190°C). Arrange a rack in the center of the oven. Line a baking sheet with parchment paper or a nonstick silicone baking mat.
Roll the dough 1/8 to 1/4 inch thick. Cut rounds slightly larger than the diameter of your ovenproof bowls or ramekins. Place the rounds on another baking sheet and set aside in the refrigerator.
Ladle the chicken mixture into the ramekins. The mixture should come all the way up to the top of the bowl so that the crust doesn’t sink into the filling and soak. Top each bowl with a round of dough, tucking the excess around the edges of the ramekins. Brush with the egg wash. Use a knife or scissors to cut a vent in the crust. Set the dishes on the prepared baking sheet and bake, rotating the sheet about two-thirds of the way through the cooking, until the crust is golden brown and the juices are bubbling, 30 to 40 minutes. Serve hot.