In a medium bowl, mix the flour and spices (rosemary through nutmeg). Toss the chicken cubes in the spiced flour mixture to coat. Set aside.
In a large skillet set over medium heat, heat 2 teaspoons of the olive oil. Add the squash, flip them around a little bit to coat them in oil, then cover the pan and let the squash cook for 10 minutes, stirring occasionally. Add the mushrooms and cook for another 5 minutes, uncovered. By this time, the mushrooms should be browned and the squash should be showing signs of softening (but doesn’t need to be softened all of the way yet). Remove the squash and mushrooms from the pan.
Add the remaining 2 teaspoons of olive oil to the pan. Once hot, add the flour and spice coated chicken cubes, shaking off excess flour before adding the chicken pieces to the pan. Sauté on one side for about 2 minutes, until a golden brown crust forms, then flip the chicken pieces over and repeat on the other side. Remove the chicken from the pan (it’s ok that it’s not fully cooked yet).
Add the cider, hard cider, and mustard to the skillet and whisk to combine, making sure to scrape up any browned bits of fond from the bottom of the pan. Simmer for 5 minutes to reduce the sauce by about 1/3.
Add the chicken, squash, and mushrooms back to the skillet with the cider mixture. Stir to coat then simmer, stirring occasionally, for 7 – 8 minutes (until the chicken pieces are cooked to 165° F).
Serve warm. This dish reheats nicely (you may need to add a bit more cider or water if the sauce gets too thick).