Place the flour, cheeses, shortening and salt into a medium bowl and with your fingertips rub in the shortening until the mixture resembles a coarse meal. Add water and mix with a fork just until the mixture comes together. Form into a ball, wrap in plastic wrap and allow to rest in refrigerator for 30 minutes before rolling out.
Microwave all of the vegetables together for about 5-6 minutes until they are just tender. Set aside.
Melt butter in medium sauce pan. Stir in flour, salt, pepper and tarragon and cook for 2 minutes over medium heat. Whisk in chicken broth and let cook, stirring frequently until mixture thickens and comes to a full boil. Add vegetables and cooked chicken.