Preheat your oven to 400°F (200°C). Position the rack in the center of the oven. Set out a 9x13-inch baking dish or a deep 10-inch oven-safe skillet.

In a large skillet or Dutch oven, melt the butter with the olive oil over medium heat. This should take about 1–2 minutes. The butter should be fully melted and starting to foam slightly.

Add the diced onion and cook for 4–5 minutes, stirring occasionally, until it turns translucent and softens. It should smell sweet and fragrant.

Add the minced garlic, diced carrots, and diced celery. Cook for another 5–6 minutes over medium heat, stirring frequently, until the carrots begin to soften slightly. Checkpoint: You are ready to move on when the vegetables have softened at the edges but still hold their shape.

Add the sliced cremini mushrooms. Cook for 4–5 minutes over medium-high heat without stirring too much, so they can brown slightly. They should shrink in size and turn golden at the edges.

Reduce the heat to medium. Sprinkle the flour evenly over all the vegetables. Stir constantly for 1–2 minutes until the flour coats everything and there are no white dry patches visible. It should smell slightly nutty.

Slowly pour in the chicken broth, about 1/2 cup at a time, stirring constantly with a wooden spoon or whisk to prevent lumps. Add the whole milk and heavy cream the same way. This will take about 2–3 minutes total.

Increase heat to medium-high and bring the mixture to a gentle simmer, stirring frequently. Cook for 4–6 minutes until the sauce thickens noticeably — it should coat the back of a spoon and leave a clear line when you drag your finger through it. Checkpoint: You are ready to move on when the filling holds its shape briefly when stirred and looks creamy and thick.

Reduce heat to low. Add the shredded cooked chicken, frozen peas, thyme, rosemary, onion powder, and garlic powder. Stir to combine everything thoroughly. Season generously with salt and black pepper. Taste and adjust seasoning. Let it warm through for 2–3 minutes.

Remove the skillet from heat and let the filling cool for 10 minutes. This prevents the puff pastry from wilting when placed on top. Checkpoint: You are ready to move on when the filling is warm but no longer steaming heavily.

If using a separate baking dish, pour the filling in now. Lightly flour your work surface. Unfold the thawed puff pastry sheet and gently roll it out slightly to fit over your dish, leaving about 1 inch of overhang on all sides.

Carefully drape the puff pastry over the filling. Press the edges down firmly against the rim of the dish to seal. Fold any excess pastry under itself and crimp with a fork for a decorative edge.

In a small bowl, beat the egg with 1 tablespoon of water. Brush the entire surface of the puff pastry evenly with the egg wash using a pastry brush. This gives it a deep golden color.

Using a sharp knife, cut 4–5 small slits (about 1 inch long) in the center of the pastry to allow steam to escape during baking. Without these vents, the pastry can become soggy underneath.

Place the dish on the center rack of the preheated 400°F oven. Bake for 25–30 minutes until the pastry is puffed up dramatically, deeply golden brown, and you can see the filling bubbling around the edges and through the vents. Checkpoint: You are ready to remove it when the top is uniformly deep golden (not pale yellow) and the filling is audibly bubbling.

Remove from the oven and let it rest on a wire rack for 8–10 minutes before serving. This allows the filling to set slightly so it does not pour out when scooped. Garnish with freshly chopped parsley and serve hot.