In a large bowl, whisk together flour, sugar and salt. Work shortening into the flour mixture with your fingertips until evenly dispersed throughout.
Work butter and herbs into the flour mixture with your fingertips, creating flat dime size pieces.
Mix ice water and vinegar together. Sprinkle over flour mixture a little at a time, stirring until dough comes together. You may not need all of the water depending on your climate.
Turn dough out onto a lightly floured surface. Knead dough briefly by folding dough over onto itself a few times. Do not over work dough! Form dough into a disk. Cover with plastic wrap and refrigerate at least one hour or up to overnight.
Filling
In an 8-inch cast iron skillet, heat butter over medium heat. Sauté onions, celery and carrots until onions are translucent, stirring occasionally. Sprinkle with flour and cook one to two more minutes, stirring frequently.
Stir in chicken broth slowly. If your sauce seems lumpy, use a whisk and vigorously whisk until smooth. Simmer on low until thickened. Stir in heavy cream. Add herbs and seasonings to taste. Add a splash of chicken broth if your sauce seems too thick.
Mix in chicken and peas. Remove form heat. Set aside until skillet is cool enough to handle.
Assembly
Preheat oven to 400ºF.
On a lightly floured surface, roll dough out to a 10-inch round. Place dough over filling, tucking ends under and crimping with your fingers or with a fork. Brush pie with egg wash. Cut a 1-inch X in the center of pie to allow steam to escape.
Bake until golden brown, about 40 minutes. Cover edges of crust with strips of foil if they start to brown too fast.