Preheat to 375 degrees.
In a 10 inch cast iron skillet melt butter and olive oil over medium heat until butter is foaming.
Add onion, celery, carrots, and ½ teaspoon of table salt. Stir well and cook until the vegetables are tender, stirring frequently, about 8 to 10 minutes.
Sprinkle over thyme and oregano and cook until fragrant, about 20 to 30 seconds.
Sprinkle over flour, stir well to coat all the vegetables and continue to cook for 2 minutes, stirring frequently.
Add chicken stock and milk and stir well. Bring up to a simmer, stirring frequently until thickened. Note: take care to keep stirring to avoid lumps and sticking.
Add chicken and peas. Stir well until chicken and peas are warmed through. Season with black pepper to taste and more salt, if needed, to taste. Smooth into an even layer. Remove from heat and set aside.
Remove puff pastry from refrigerator and on lightly floured surface cut into about 12 rectangles, about 1 ½ x 3 inches each. Note: This doesn't have to be perfect.
Layer puff pastry rectangles on top of chicken mixture in skillet, overlapping is okay to ensure it's covered.
Make egg wash: Whisk together 1 egg and 1 teaspoon of water. Brush egg wash gently on top of puff pastry.
Place skillet on baking sheet and immediately transfer to the oven.
Bake until the pastry is golden brown and slightly puffed, about 20 to 30 minutes.
Cool for at least 10 minutes (it's piping hot y'all!) and serve. It pairs lovely with a light green salad.