Preheat oven to 425°. In large pot, combine olive oil, onions, and garlic over medium-high heat.
Sauté until onions start to turn translucent, about 1 minute.
Add carrots, celery, and corn. Cook until softened, about 8 minutes.
Stir in chicken, then sprinkle flour over the top and stir until combined with vegetables.
Pour in chicken broth and stir until thickened, about 5 minutes.
Season well with salt, pepper, thyme, and parsley to taste.
Add half & half, then stir and let bubble up and thicken, about 3 minutes.
Pour filling into a 9-inch pie dish.
Roll pie crust into a 10-inch diameter circle and place over pie filling.
Use a sharp knife to cut 5-6 small slits in the top to allow steam to escape.
Whisk egg and 2 Tbsp. water together, and brush over the surface of crust.
This egg wash will give the pie a nice golden-brown color.
Place pie in oven for 30-35 minutes.
Once out of the oven, rest for 15 minutes to cool slightly before slicing.
