In a medium skillet, heat olive oil over medium-high heat. Add onion, carrot, and mushroom, and sauté several minutes until tender. Add garlic, thyme, and flour and cook an additional 1-2 minutes.
Reduce heat to medium and gradually begin to drizzle in chicken broth, stirring constantly until smooth. Add milk and heavy cream to skillet and bring to a low simmer over medium heat, stirring occasionally until sauce is thickened.
Remove sauce from heat and stir in turkey and wild rice. Season with salt and pepper to taste and pour into the bottom of an 9-inch cast iron skillet or baking dish. Set aside for a moment.
Preheat oven to 400F. Remove pastry from fridge and turn onto a floured surface. Roll pastry out into a 9-inch square and cut into 9 pieces.
Place dough squares on a large parchment paper-lined sheet pan and brush with egg white-water mixture. Bake squares at 400F 10-15 minutes until a very light golden brown.
Carefully transfer dough squares to skillet and arrange on top of filling. Bake pot pie at 400F an additional 20-25 minutes until filling is bubbly and pastry is golden brown. Allow pot pie to cool several minutes then serve warm. Enjoy!