For the crust: Whisk together the flour and salt in a large mixing bowl. Add the butter and cream cheese and toss to coat with the flour. Using a pastry blender, two knives or your fingertips, work the fat into the flour until the mixture looks like moist cornmeal. Sprinkle in the water, a tablespoon at a time, gently working the mixture into a shaggy dough that just comes together when pinched. Gather the dough into a ball, then divide it in half and form into two disks. Wrap each disk in plastic and refrigerate for at least 20 minutes.
For the filling: Heat the olive oil in a large skillet over medium heat. Add the onions and garlic and sauté until translucent, about 4 minutes. Add the potatoes and continue to cook, stirring occasionally, until the potatoes have softened, about 10 minutes. Add the green beans and bell peppers; cook for another 2 minutes. Transfer to a large bowl with the chicken pieces and set aside.
In a saucepan, melt the butter over medium heat and whisk in the flour. Continue whisking over low heat for 2 to 3 minutes (it's OK if the roux colors -- I like the nutty flavor), and then add the cream, milk and vermouth. Continue stirring over medium heat until the mixture thickens, about 15 minutes. Remove from the heat and stir in the Parmesan, thyme and lemon zest. Season with salt and pepper, then pour the roux over the chicken-vegetable mixture. Stir to combine. Cover with plastic wrap and refrigerate until cool.
Roll out one of the disks of dough into a rough 12-inch round. Line a 9-inch pie plate with the dough so that it overhangs the edge by about 1/2 inch. Dock the dough in the pie pan by pricking the bottom with the tines of a fork. Add the filling to the lined pie plate. Roll out the second disk of dough into a 10-inch round and top the pie. Crimp the edges to seal. Cut slits in the top of the pie to vent. Place in the freezer until the crust feels frozen, 10 to 15 minutes. (At this point, the pie can be wrapped in plastic and then foil and frozen for up to a month. See Cook's Note for baking instructions.)
Preheat the oven to 375 degrees F. Brush the pot pie with egg wash, place on a baking sheet and transfer to the oven. Bake until the crust is golden brown and the filling is bubbling, about 40 minutes.
Freeze the cubed butter and cream cheese for 5 to 10 minutes before making the crust. You can substitute white wine for the vermouth, though I prefer the flavor of vermouth for this recipe. If baking the pot pie directly from frozen, unwrap and increase the baking time by about 30 minutes; cover with foil if it's browning to quickly.