Preheat your oven to 400°F.

Melt the butter in a large Dutch oven over medium heat. Add the diced onion, carrots, and celery. Cook for 5–7 minutes until softened. Stir in the garlic and cook for 1 more minute.

Add the raw chicken pieces to the pot. Season with salt, pepper, and Italian seasoning. Cook for about 5 minutes, stirring occasionally, until the chicken is just starting to brown.

Sprinkle the flour over the chicken and vegetables. Stir to coat evenly and cook for 1–2 minutes.

Gradually pour in the chicken broth while stirring, scraping up any bits from the bottom. Add the milk and stir well to combine.

Bring the mixture to a gentle simmer. Cook for 5–7 minutes, stirring often, until the sauce thickens.

Stir in the frozen peas and adjust seasoning with additional salt and pepper if needed. Remove from heat.

Roll out the puff pastry slightly if needed to fit over the Dutch oven. Place it carefully over the filling. Trim any excess, and gently poke a few holes in the puff pastry with a knife or fork to let steam escape.

Brush the puff pastry with the beaten egg.

Bake uncovered for 20–25 minutes, or until the pastry is puffed and golden brown.

Let cool for 5–10 minutes before serving.