Heat the butter in a large sauté pan or pot over medium heat. When it's melted, add the onion, carrots, and garlic and cook until the onion is translucent and the carrots begin to soften, about 5 minutes. Add the mushrooms and pearl onions and cook, stirring occasionally, for another 5 minutes. Stir in the chicken and the flour, using a wooden spoon to ensure the vegetables and meat are evenly coated with flour. Slowly pour in the chicken broth, using a whisk to beat it in to help avoid clumping with the flour (having the broth warm or hot helps smooth out the sauce). Once the broth is incorporated, add the milk and half-and-half and simmer for 10 to 15 minutes, until the sauce has thickened substantially and lightly clings to the vegetables and chicken. Stir in the peas. Season with salt and pepper. Preheat the oven to 375°F. Cut the pastry into quarters. Roll out each piece on a floured surface to make a 6" square. Divide the chicken mixture among 4 ovenproof bowls. Place a pastry square over the top of each bowl, and trim away the excess with a paring knife; pinch the dough around the edges of the bowl to secure it. Brush the tops with the egg whites and bake until golden brown, about 25 minutes.