In a large skillet, heat olive oil over medium heat. Add carrots, green beans, and onion and cook until they begin to soften, about 3 minutes.
Add garlic, thyme, and basil. Continue cooking until soft, 3–4 minutes more.
Add chicken and combine all items in the skillet very well.
Sprinkle in flour and stir to coat well. Slowly pour in chicken broth, stirring as you pour. Bring to a boil, and sauce will begin to thicken. The mixture should be somewhat loose, not overly thick. If the mixture is too thick, it will be dry after baking.
Place one pie crust on the bottom of a deep-dish pie plate. Pour the chicken mixture into the pie crust . Top the mixture with the other pie crust.
Seal the edges and brush with the egg wash. Cut small slits into the top to allow steam to escape while baking.
Place pie on a baking sheet and bake until golden brown, 35–40 minutes.
Remove from oven and allow to rest for about 10 minutes before serving.