Preheat the oven to 400 degrees F.
Heat a large over-safe (preferably cast iron!) skillet over medium heat. Add 1 tablespoon of olive oil and butter each. Toss in the shallots and butternut squash with a pinch of salt and pepper. Stir in the cardamom and cinnamon. Cook, stirring occasionally, for about 5 to 6 minutes. Add in the mushrooms and stir to toss. Cook for another 5 minutes, until the mushrooms slightly soften. Remove the shallots, squash and mushrooms from the skillet and place them in a large bowl off to the side.
Increase the heat until the skillet to medium-high. Stir together the 1/2 cup flour, salt, pepper, garlic, paprika and nutmeg in a bowl. Add the remaining oil and butter to the hot skillet. Take each piece of chicken and dredge it lightly in the seasoned flour then place it in the skillet. Brown the chicken on all sides until deeply golden. Once finished, remove the chicken from the skillet and place it on a plate.
With the heat still on medium-high, slowly pour in the hard cider which scraping the bottom of the skillet to remove the brown bits of flavor. Stir in the apple cider. Sprinkle in a tablespoon of flour and the dijon mustard, whisking until combined. Cook the mixture until it is bubbling and reduces by just a third. Reduce the heat to low and stir in the cream. Place the chicken back in the skillet and cover with the shallots, butternut and mushrooms. Place a few rosemary and thyme sprigs on top. Cover the skillet and place it in the oven. Braise for 25 to 30 minutes. Let cool slightly before serving.