Adjust oven rack to the middle position and preheat to 400°F (205ºC).
Heat oil in a 10-inch cast iron skillet over medium heat until it just shimmers. Add the chicken and cook, stirring occasionally, until cooked through, about 6 minutes.
Use a slotted spoon to transfer the chicken to a plate, leaving the tasty bits behind.
Build the brown-butter vegetable base, then make the roux
Melt butter in the same skillet over medium, then let it cook until the milk solids turn golden and smell nutty.
Stir in carrots, celery, leek, garlic, thyme, salt, and cayenne. Cook, stirring often, until the vegetables are tender, about 6 minutes.
Sprinkle with flour and stir constantly until everything is coated and the flour begins to stick to the skillet, about 1 minute.
Simmer into a creamy filling and finish
Stir in broth and 1/4 cup of the cream. Bring to a simmer over medium and cook, stirring often, until thickened, about 3 minutes.
Stir in peas, lemon zest, and the reserved chicken, then remove from heat.
Top, vent, and bake until bubbly
Lay puff pastry over the filling and tuck the edges inside the skillet so the mixture is fully covered.
Whisk egg with the remaining 1 teaspoon heavy cream, then brush the pastry evenly with the egg wash.
Cut a 3-inch “X” in the center of the pastry so steam can escape.
Set the skillet on a rimmed baking sheet and bake at 400°F until the top is golden and the filling is bubbling, about 30 minutes.
Let stand 10 minutes before serving so the sauce sets up.
Notes
For the best texture, assemble with puff pastry and bake promptly; venting with a 3-inch “X” helps prevent gummy pastry.Nutrition information is estimated based on common ingredients and serving sizes and may vary.