Preheat oven to 400 degrees.
Bring chicken and broth to a simmer in a large skillet over medium heat. Cover and simmer until chicken is just done, about 10 minutes.
Transfer chicken to a large bowl, reserving chicken stock.
Heat the olive oil, over medium-high heat, in the large skillet. Add the onions, carrots and celery and sauté until tender, about 5 minutes.
Add the kale and lemon zest. Sauté one minute. Season with salt and pepper.
While the vegetables are sautéing, shred the chicken. Transfer the vegetables to the bowl with the chicken and set aside.
Heat butter, over medium heat, in the skillet. When melted, add the flour and whisk until combined.
Whisk in the chicken stock, buttermilk, lemon juice and thyme. Bring to a simmer and continue to whisk until it begins to thicken, about 1 minute. Season with salt and pepper.
Pour the sauce over the chicken and vegetables and mix to combine.
Add the peas and parsley and mix well.
Pour the chicken mixture into a large pie plate or casserole dish. Lay the puff pastry dough over the filling, covering completely and pressing the excess around the edges. Cut a vent hole in the center.
Bake for 25 - 30 minutes or until the puff pastry is golden brown and the filling in bubbly.
*You can also substitute a cooked, rotisserie chicken.