Using a medium saucepan, place the chicken breasts, chicken broth, salt and pepper. Bring to a boil and reduce to low heat. Cover and simmer for 15-20 minutes, until chicken is almost cooked through.
Meanwhile, in a small sauce pan add the carrots and potatoes, cover with water. Boil, reduce heat and cook until about half done. Blanch green beans. Remove vegi's from water into a bowl.
Now the chicken should be ready. Remove the chicken and let cool. Cut into cubes. Pour 1 cup of the chicken broth into a measuring cup. Allow to cool while you're doing the vegi's. Once the vegi's are done combine the cream with the 1 cup of broth.
In the same pan the chicken was cooked in, melt butter over medium heat. Add the onion and celery. Saute, stirring, for 3 minutes. Stir in flour until well blended.
Gradually stir in broth mixture and the herbs. Simmer, stirring constantly, until the sauce thickens and boils. Add the chicken and vegetables to the broth and stir to combine.
Line one deep dish pie plate with one of the pie crusts and ladle the chicken filling into it. filling it almost to the top. (Or prepare your mini pie tins). I always make my pie crust dough ahead of time and have balls of dough in the freezer. I let them thaw for about 45 minutes before rolling them out. I prefer to make mini pies in jumbo muffin tins, or large ramekins.
Cover mixture with second pastry, making sure to seal the top & bottom crusts by pinching them together. This will keep the filling from leaking out. Cut a couple of vent holes in the pastry top. Cut foil strips to wrap the edges of the pie to prevent buring.
Bake at 400 degrees for 25 minutes, then remove the foil strips and bake for an additional 5 minutes or until the crust is golden brown. The filling will be very hot! Let cool for 10 minutes and serve.