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. (This allows steam to escape as the pie bakes.)

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.