Preheat oven to 400°F. Season chicken breasts and thighs with kosher salt and ground black pepper on both sides.

Heat olive oil in a large 12-inch cast iron skillet over medium-high heat. Cook chicken for 4-5 minutes per side until golden. Remove to a cutting board, slice into strips, and set aside.

Add butter, chopped carrots, onions, and sliced mushrooms to the skillet. Sauté until vegetables are tender, about 6-7 minutes.

Reduce heat to medium and add garlic, thyme, salt, pepper, and flour. Stir to combine and cook for about 30 seconds.

While stirring, slowly add chicken stock to combine. Add cream and simmer until thickened, about 3-5 minutes.

Remove from heat. Stir in frozen peas and reserved sliced chicken. Set skillet aside to cool slightly while you work on the puff pastry.

Lay out each sheet of puff pastry on a cutting board and cut into ½-inch strips. Twist each strip and arrange in a circular pattern from the outside of the skillet to the inside, overlapping pieces so they bake together. Continue to the center until surface is covered.

Use a pastry brush to lightly brush whisked egg over the top of the puff pastry.

Bake in the oven for 35-40 minutes until the puff pastry is golden brown. Remove and allow to sit for 5-10 minutes before serving.