Put the chicken in a large stockpot and cover with 1 gallon of cool water. Add the vegetables and herbs and bring the pot up to a boil over medium-high heat. Skim well; then simmer, uncovered, for 45 minutes, skimming frequently as the oil rises to the surface. What we're doing here is not only cooking the chicken but also creating the base sauce for the pot pie. Remove the chicken to a platter to cool. Continue to cook down the chicken broth for another 15 minutes to condense the flavor; you should have about 8 cups when you're finished. Using a colander, strain the chicken broth into another pot and discard the solids. When the chicken is cool enough to handle, shred the meat and discard the skin and bones.
Wipe out the stockpot and put it back on the stovetop over medium heat. Melt the butter and then whisk in the flour to form a paste. This is a roux, which will act as a thickener. Now, gradually pour in 8 cups of the chicken broth, whisking the entire time to prevent lumps. Whisk and simmer for 10 minutes to cook out the starchy taste of the flour and thicken the broth; it should look like cream of chicken soup. Season with salt and pepper. Fold in the shredded chicken, carrots, pearl onions, peas, and parsley. Stir to combine and turn off the heat.
Preheat the oven to 350 degrees F. Lay the thawed puff pastry sheets on a lightly floured, cool surface. Invert individual crocks on the pastry sheet and, using a sharp knife, cut circles around the outside of the bowls, slightly larger than the bowl itself. Fill the crocks 3/4 of the way with the chicken mixture, making sure each serving has a nice amount of chicken, vegetables, and broth. Carefully cap each crock with a pastry circle, pressing the dough around the rim to form a seal. Lightly beat the egg with 3 tablespoons of water to make an egg wash and brush some on the pastry. Sprinkle the pastry with the Parmigiano. Set the crocks on a cookie sheet and transfer to the oven. Bake for 20 minutes, until puffed and golden.
Notes: If using fresh pearl onions, blanch for 2 minutes in salted boiling water; then pinch the skins off -- if using frozen, run under cool water for 2 minutes to thaw.
If using fresh peas, blanch for 2 minutes in salted boiling water--if using frozen, run under cool water for 2 minutes to thaw.