Preheat the oven: Set to 400°F (200°C). If using puff pastry, let it thaw according to package directions.
Sauté the veggies: Warm olive oil in a large skillet over medium heat.
Add onion, carrots, and celery with a pinch of salt. Cook 5–6 minutes until softened. Stir in mushrooms and cook 3–4 minutes, then add garlic for 30 seconds.
Build the roux: Sprinkle flour over the vegetables.
Stir for 1–2 minutes until the flour is fully moistened and slightly toasty. This step thickens your sauce later.
Add liquids: Slowly whisk in chicken broth, scraping up any browned bits. Add milk.
Bring to a gentle simmer and cook 4–5 minutes until thick and creamy.
Season and enrich: Stir in thyme, rosemary, paprika, and plenty of black pepper. Taste and add salt as needed.
Fold in the good stuff: Turn off the heat. Stir in Greek yogurt until smooth, then add chicken, peas, and parsley.
The filling should be thick but spoonable.
Assemble the pie: Transfer filling to a 9-inch pie dish or an 8x8-inch baking dish. Lay the puff pastry or pie crust over the top. Trim or tuck the edges, then cut a few small slits for steam.
Finish the top: Brush with egg wash for shine if you like.
For extra texture, sprinkle a pinch of flaky salt or cracked pepper on the crust.
Bake: Place on a baking sheet to catch drips. Bake 22–28 minutes, until the crust is puffed and deeply golden and the filling is bubbling at the edges.
Rest, then serve: Let it sit 10 minutes. This helps the filling set so slices hold together.
Serve warm.