Sauté the mire poix. Preheat an oven to 425°F/220°C. Heat 2 Tbsp olive oil and 2 Tbsp butter in a large skillet or braiser over medium-high heat. Add 2 cups diced onion, 2 cups diced carrots, and 1 cup chopped celery. Season with 1 tsp Kosher salt, then cook, stirring occasionally, until softened, about 4-6 minutes. Add 4 chopped garlic cloves, then cook 1 minute more.
Add the herbs and spices. Add 2 Tbsp chopped sage, 1 Tbsp chopped rosemary, 1 Tbsp chopped thyme, 1 tsp sweet paprika, ½ tsp black pepper, and ¼ tsp curry powder (optional) then stir and cook for 1 minute.
Stir in the flour. Add ⅓ cup all-purpose flour to the pan, then cook for 1 minute, stirring the whole time. The flour should absorb any liquid in the skillet.
Add liquid. Slowly whisk in 2 cups chicken broth, being sure to scrape any browned bits off the bottom of the pan. Stir in ½ cup heavy cream, then whisk well. Bring the mixture to a simmer until just barely bubbling, then add the pulled rotisserie chicken, 2 cups parboiled potatoes, and 1 cup frozen peas. Give everything a good stir, then turn off the heat.
Add the puff pastry. You can cut the puff pastry into a circle to match the size of the pan, but note that it will shrink up a bit as it cooks. Instead, we like to either make a lattice topping or cut the puff pastry into smaller 2x2-inch squares, then overlap them to cover the entire top of the skillet.
Make the egg wash. In a small bowl, whisk together 1 large egg and 1 tsp heavy cream or milk. Brush this on top of all exposed puff pastry. Sprinkle 2-3 Tbsp grated Parmesan on top along with a little freshly ground black pepper (as much as you like!), and a generous pinch or two of flaky sea salt.
Bake. Transfer to the oven and bake for 30-35 minutes, or until the puff pastry is golden brown and crispy.