Preheat oven to 400°F (205°C). Melt 4 tablespoons butter in a 10 or 12-inch heavy oven-safe cast iron skillet over medium heat. Add diced yellow onion, carrots, and celery with 1/2 teaspoon kosher salt. Sauté for 6 to 8 minutes until vegetables soften and begin to turn translucent. Stir in minced garlic and cook for 60 seconds until fragrant.

Sprinkle the all-purpose flour evenly over the softened vegetables. Cook, stirring constantly with a wooden spoon for 2 minutes to cook out the raw flour taste and coat every vegetable piece evenly.

Gradually pour in the chicken bone broth and heavy cream in a steady stream while whisking continuously to prevent lumps. Bring to a gentle simmer for 3 to 4 minutes until the velouté thickens into a glossy, velvety gravy. Stir in shredded chicken, green peas, chopped thyme, rosemary, remaining salt, and cracked black pepper. Remove skillet from heat.

Unroll the cold puff pastry sheet over the hot chicken filling in the skillet, gently pressing and tucking the overhang along the inside rim. Using a sharp paring knife, lightly score a decorative crosshatch pattern on top and cut 3 to 4 small steam slits in the center. Brush the entire pastry surface with beaten egg wash and sprinkle with flaky sea salt and fresh thyme leaves.

Transfer the skillet to the middle oven rack and bake at 400°F (205°C) for 18 to 22 minutes, until the puff pastry crust is deeply golden-brown, puffed into flaky buttery layers, and the savory gravy is bubbling gently through the vents. Let rest for 5 minutes before scooping into hearty portions.