Preheat the oven to 425 degrees F (220 degrees C). Place a rack in the upper third of the oven.
Heat oil in an 8-inch cast iron skillet over medium-high heat. Add mushrooms and onions; cook, stirring occasionally, until starting to brown, about 4 minutes. Reduce heat to medium-low. Add garlic and sage; cook, stirring occasionally, until garlic starts to brown, about 2 minutes.
Add kale and season with brown sugar, pepper, Cajun seasoning, and cardamom. Cook, tossing often, until kale is wilted, about 4 minutes. Sprinkle flour on top; cook and stir for 4 minutes.
Meanwhile, combine chicken broth, apple juice, vinegar, pepper jelly, yeast extract, and hot sauce in a small saucepan. Bring to a boil and remove from heat.
Stir broth mixture into the skillet 1/2 cup at a time. Add butternut squash and chicken. Bring to a boil. Reduce heat and simmer until squash softens and broth thickens, about 10 minutes.
Unroll puff pastry sheet and smooth out any creases. Drape pastry over the skillet, allowing edges to hang over the sides. Whisk egg and water together in a small bowl. Brush egg wash over pastry. Cut four 1-inch slits in the top to vent.
Bake in the preheated oven until pastry is golden brown and crisp, about 20 minutes. Let cool for 10 minutes before serving.