Using a food processor, pulse together the flour and salt until combined. Add butter and pulse until mixture becomes coarse meal, about 15 pulses.
In a small bowl, combine the eggs and water. Add the egg mixture to food processor and pulse until it starts to clump together. DO NOT pulse until a ball is formed. It should be crumbly with large clumps of dough.
Turn dough only a lightly floured surface and form into a ball. Flatten slightly to form a disk before dividing into 2 discs. Wrap each in plastic wrap and put in fridge for at least 1 hour or up to 3 days.
Melt the butter over medium heat in a large oven-safe pan or Dutch oven. Add diced onion and poultry blend and sauté for 5 minutes. Add the carrots and celery, cooking for another 5 minutes. Stir garlic in and cook for 30 seconds. Add the flour and cook for about 1 minute.
Slowly add the chicken stock and milk, stirring, until the mixture begins to bubble and thickens slightly, about 4-5 minutes. Add the chicken, ham, peas, lemon juice and salt. Taste for seasoning and adjust if needed. Set aside
Remove one disk of dough from the fridge about 10 minutes before you begin rolling. On a lightly floured surface, roll the dough so that it is about 3 inches longer than you pie dish and is between ⅛ inch and ¼ inch in thickness. Press the dough into the bottom and along the sides of the pie plate, being careful not to puncture the dough. Use kitchen scissors to trim the dough overhang to about 1 ½ inches.
Arrange a rack in the middle of the oven, set a baking sheet on it, and heat the oven to 375°F. Whisk the egg and water together to make an egg wash. Brush the inside of the pie crust with a thin layer of the egg wash. Pour the filling into the crust.
Roll out the second pie dough so that it is just larger than the pie dish. Add over the filling and seal and flute the edges. Cut 4 slits in the top crust. Brush with egg wash.
Place the pie on the hot baking sheet and bake until the crust is golden brown, about 50 to 60 minutes. If the edges of the pie crust are browning too quickly, cover the edges of the pie with aluminum foil. Let cool for 15 minutes before serving.