Preheat the oven to 375°F (190°C) and lightly grease a 9-inch deep-dish pie plate or an oven-safe skillet.
In a large skillet over medium heat, melt 4 tablespoons of butter. Add the sliced leeks and sauté for 4 to 5 minutes until softened.
Stir in the all-purpose flour and cook for 1 minute, stirring constantly to form a roux.
Gradually whisk in the chicken broth and heavy cream. Bring the mixture to a gentle simmer and cook for 3 to 4 minutes until the sauce thickens.
Remove the skillet from the heat. Stir in the shredded chicken, green peas, chopped spinach, fresh dill, Dijon mustard, kosher salt, and black pepper. Pour the filling into the prepared pie plate.
Layer the thawed phyllo sheets one at a time over the filling. Lightly brush each sheet with melted butter as you layer, crinkling and folding the edges loosely to fit the dish. Do not press them flat; the wrinkles create the flakiest texture.
Bake for 25 to 30 minutes, or until the phyllo crust is deeply golden brown and the filling is bubbling around the edges.
Let the pot pie cool for 10 minutes before serving to allow the creamy sauce to set.