Into a mixer, add the All-Purpose Flour (1/3 cup), Granulated Sugar (1 tablespoon), Butter (1/2 cup) and Salt (1/2 teaspoon). Add a tablespoon of Water (3 tablespoon) at a time, and mix everything until combined.
Wrap around and press the dough with a plastic rubber, and put it in the fridge for about 30 minutes.
Into a cooking pan, add the Extra-Virgin Olive Oil (2 tablespoon), Butter (1/4 cup) Onion (1 cup), Carrot (2/3 cup), Celery (2/3 cup), Salt (1/2 teaspoon), and Freshly Ground Black Pepper (to taste). Give the mixture a gentle stir and cook for about 5 minutes.
After 5 minutes, add the Chicken Stock (1/2 cup), Dried Thyme (1/2 teaspoon), and Dried Parsley (1 teaspoon). Add All-Purpose Flour (4 tablespoon) and give it a gentle stir and let it simmer for 5 minutes.
Then add the Heavy Cream (1/2 cup), Green Peas (3/4 cup), and Chicken (2 cup). Give it a good stir, and cook for 10 minutes.
Adjust seasoning with Salt (1/2 teaspoon) and Freshly Ground Black Pepper (to taste). And set aside while you work on the pie crust.
On a floured surface, roll our your dough. The fill in the pie dish with the filling mixture, and smoothly cover the surface by the flat dough. Make sure to tug in the edges. Have some vent holes at the center, and bake under 400 degrees F (200 degrees C) for about 40 minutes.