Preheat oven to 425°.
On a floured surface, roll out pie crust as needed so that it is slightly wider than a 9- or 10-inch cast-iron skillet. Transfer dough to a sheet of parchment paper. Place skillet top-down on the dough. Cut around the edges of the pan with a paring knife. Remove the dough from around the outside of the pan, then remove skillet. Place the parchment with remaining dough on a sheet pan. Refrigerate until ready to use.
In the cast iron skillet, heat olive oil over medium heat. Add mushrooms, onion, chard stems (but not leaves yet), chicken thighs, rosemary, thyme, ½ tsp salt, and ¼ tsp pepper. Cook for 5-7 minutes, stirring occasionally, until mushrooms and chicken are browning and onions are softened and turning translucent.
Gradually add chard greens to the pan and stir until wilted. Mix in carrots, potatoes, and flour. Cook for 1-2 minutes, stirring occasionally.
Mix in sherry, broth, and cream. Bring to a gentle simmer.
Remove skillet from heat. Remove parchment from pie dough and place pie dough on top of the filling in the skillet. Cut a circle out of the middle of the dough with a paring knife and cut several slits radiating towards the outside of the pan.
In a small bowl beat together an egg with 1 tbsp water. Brush egg mixture on top of crust.
Place skillet on a sheet pan and bake for 25-30 minutes, or until crust is deeply golden brown. Allow to cool for 10 minutes before serving.