Preheat and prep: Heat your oven to 375°F (190°C). Grease a deep 9-inch pie dish or a 2-quart baking dish.
Sauté the vegetables: In a large skillet, melt butter with olive oil over medium heat.
Add onion, celery, and carrots (or rutabaga). Cook 5–7 minutes until softened. Stir in cauliflower and cook 3–4 more minutes.
Add garlic and cook 30 seconds.
Season and add chicken: Stir in thyme, parsley, rosemary, salt, and pepper. Add the cooked chicken to the skillet and toss to coat.
Make the creamy sauce: Pour in chicken broth and heavy cream. Add cream cheese and stir until melted and smooth.
Simmer 2–3 minutes to thicken slightly. Stir in Parmesan. Taste and adjust seasoning.
Transfer the filling: Pour the mixture into the prepared pie or baking dish and smooth the top.
Make the crust dough: In a mixing bowl, whisk almond flour, coconut flour, baking powder, and salt.
Cut in the cold butter with a pastry cutter or your fingertips until the mixture looks like coarse crumbs.
Bind and chill: Whisk the egg and vinegar together, then add to the flour mixture. Add cold water 1 tablespoon at a time until a soft dough forms. It should hold together but not be sticky.
Form into a disk, wrap, and chill 10–15 minutes for easier rolling.
Roll the crust: Place the dough between two sheets of parchment. Roll into a circle slightly larger than your dish. Peel off the top parchment.
Cover the pie: Flip the crust onto the filling and peel off the second parchment.
Tuck or trim edges as needed. Cut 3–4 small slits for steam. Brush with beaten egg for shine, if using.
Bake: Place on a baking sheet to catch drips.
Bake 25–30 minutes until the crust is golden and the filling is bubbling at the edges. If browning too fast, tent loosely with foil.
Rest and serve: Let the pot pie rest 10–15 minutes before slicing. This helps the filling set and makes for cleaner servings.