Cook the aromatics: Melt the butter in a large skillet over medium heat. Add the shallot and cook for 2 to 3 minutes until softened. Stir in 1/4 teaspoon salt. Add the garlic and cook for about 30 seconds, just until fragrant. Do not let it brown.
Make the roux: Whisk in the flour and cook for 2 to 3 minutes, stirring constantly, until the mixture smells slightly nutty and no longer tastes raw.
Add the liquids: Slowly whisk in the chicken stock, then the milk, then the heavy cream. Simmer for several minutes, whisking often, until the sauce thickens and coats the back of a spoon.
Add the cheese: Reduce the heat to low or remove the skillet from the heat. Stir in the Parmesan cheese a little at a time until smooth.
Season the sauce: Stir in the black pepper, Italian seasoning, and nutmeg. Taste and add the remaining salt as needed.
Add the filling ingredients: Fold in the shredded chicken, broccoli, carrots, and peas. Cook for 2 to 3 minutes until the filling is thick, creamy, and fully combined.
Finish the filling: Remove from the heat and stir in the lemon juice. Taste and adjust black pepper if needed.
Prepare the crust: If using a bottom crust, blind bake it for 8 minutes at 400°F, or add a light cheese or breadcrumb barrier to help prevent sogginess.
Assemble the pot pie: Spoon the filling into a 9-inch pie dish, leaving a little room at the top. Add the top crust, seal the edges firmly, cut 4 to 5 vents, and brush with the beaten egg.
Bake: Bake at 400°F for 30 to 35 minutes, or until the crust is golden and the center is visibly bubbling.
Cool before serving: Let the pot pie rest for 15 to 20 minutes before slicing and serving.