Special equipment: kitchen string
To make the pastry: Combine the flour and salt in a very large mixing bowl. Add the butter and mix with a pastry blender or your hands until the mixture resembles coarse crumbs. Pour in the ice water and work it in to bind the dough until it holds together without being too wet or sticky. Squeeze a small amount together, if it is crumbly, add more ice water, 1 tablespoon at a time. Wrap the large ball of dough in plastic wrap and refrigerate while preparing the rest of the recipe (feel free to make the dough the night before if you prefer.)
To make the chicken broth: Put the chicken in a large stockpot and cover with 3 quarts of cool water. Add the vegetables and herbs and bring it up to a boil over medium-high heat. Simmer, uncovered, for 1 hour; 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. It's the old trick of killing two birds with one stone -- or one pot as the case may be. Remove the chicken to a platter to cool. When it's cool enough to handle shred the chicken meat, discarding the skin and bones, and set aside. Using a colander, strain the chicken broth into another pot and discard the solids; you should have about 2 quarts when you're done.
To make the pot pie filling: Wipe out the stockpot and put it back on the stovetop over medium-low heat. Melt the butter and just as the foam subsides, add the flour, stirring constantly with a wooden spoon or whisk to prevent lumps. This is a roux, which will act as a thickener. Cook and stir the roux until it's the color of a California blonde. Now, gradually pour in the reserved 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. Fold in the carrots, pearl onions, peas, garlic, rosemary, thyme, and lemon juice; stir to combine. Simmer for 10 minutes to soften the vegetables a bit; season the mixture with salt and pepper. Stir in the shredded chicken until incorporated, remove from the heat and cool to room temperature; it will get quite thick as it cools down.
Preheat the oven to 400 degrees F.
Take the dough out of the refrigerator and set it out on the counter for 5 minutes to warm up a bit, making it easier to roll out.
Sprinkle the counter and a rolling pin lightly with flour. Roll the dough out into a big 14 by 20-inch rectangle. Carefully transfer the dough to a large rectangular pizza stone or standard sheet pan, leaving about 1 foot of the pastry hanging off 1 end. Spoon the cooled chicken filling on the pastry that is on the pizza stone, leaving a border around the sides. Lightly beat the egg with 3 tablespoons of water to make an egg wash and brush it on the dough border. Fold the "free" dough over the filling like a blanket and crimp the ends with a fork to form a tight seal; it should look like a big pillow. Brush the top of the giant chicken pie with the remaining egg wash, sprinkle with the coarse salt, and cut tiny slashes in the top so steam can escape. Place the chicken pie in the oven and bake for 45 minutes to 1 hour until puffed and golden. Let the chicken pie rest for 10 minutes before moving it. Run a spatula underneath the pot pie to loosen it from the pizza stone. Scoop out the chicken pie with a large serving spoon.