Add butter to a large saucepan over medium high heat. Once melted and foamy, add the leeks and thyme and sauté until leeks are softened, about 4-5 minutes. Season the mixture with salt and pepper and stir in the flour. Cook, while stirring, until golden blonde and pasty, about 3 minutes.
Whisk in the chicken broth along with half and half. Stir in the potatoes, carrots, and parsnips, bring to a boil, reduce heat, and simmer stirring occasionally, until sauce is thickened and the vegetables are just barely tender, about 10 minutes.
Once sauce is thick, stir in the chicken and peas. Taste for seasoning and adjust as necessary with salt and pepper. Add the mixture to a deep-dish pie plate.
Dust your work surface with flour and carefully unfold the puff pastry. Roll out to a 12x12 inch square. Drape the puff pastry over the filling and fold and crimp the pastry around the edges to make a crust. Brush with beaten egg and use a paring knife to make a few vents in the top of the crust. Place on a sheet tray and bake for 45 minutes, until puffed, golden brown and bubbling. Let rest for a few minutes before serving.