Adjust oven rack to middle position and preheat oven to 400°F (205ºC).
Heat oil in a 10-inch cast iron skillet over medium heat until just shimmering.
Add chicken; cook, stirring occasionally, until cooked through, about 6 minutes.
Using a slotted spoon, remove chicken from skillet, and transfer to a plate.
Melt butter in same skillet over medium.
Stir in carrots, celery, leek, garlic, thyme, salt, and cayenne; cook, stirring often, until vegetables are tender, about 6 minutes.
Sprinkle with flour; cook, stirring constantly, until vegetables are evenly coated and flour begins to stick to bottom of skillet, about 1 minute.
Stir in broth and ¼ cup of the cream; bring to a simmer over medium heat and cook, stirring often, until mixture thickens, about 3 minutes.
Stir in peas, lemon zest, and reserved chicken; remove from heat.
Place puff pastry on top of chicken mixture in skillet; tuck edges inside skillet so mixture is covered completely.
In a small bowl, whisk together egg and remaining 1 teaspoon heavy cream.
Using a pastry brush, brush top of pastry with egg wash.
Using a sharp paring knife, cut an 'X' about 3 inches across in middle of pastry.
Set skillet on top of a rimmed baking sheet and bake until top is golden brown and filling is bubbly, about 30 minutes.
Remove from oven and let stand 10 minutes before serving.
