In a medium fry pan saute the chicken until cooked through in olive oil. Salt and pepper while cooking. Remove from pan and set aside.
In the same pan, melt the butter and saute the onion, carrots, celery, potato, garlic, thyme, parsley, and season with salt and pepper. Continue cooking and stirring until veggies are soft.
Pour in 1 cup chicken broth and stir. Add the flour and continue stirring until the mixture thickens. Cook for 1 min. Pour in the remaining broth while stirring and bring the mixture to a boil.
Return the chicken to the pan and add the corn and peas. Bring the mixture up to a simmer and continue to cook for 15 min.
Preheat oven to 375F. Pour the mixture into four medium-sized ramekins. Roll out the puff pastry dividing one sheet for two pies, top the bowls, and trim.
Mix the egg with the water and brush the tops of the pies. Cut vents with a sharp knife.
Place bowls on a baking sheet and bake in the oven until the top is golden brown and the filling is bubbling, about 25-30 min.
Take out of the oven and let cool as the inside will be very hot.