For the polenta: Heat the olive oil in a small skillet over medium heat. Add the sage to the oil and cook until the sage is slightly crisped, about 1 minute. Remove from the heat and set aside. Meanwhile, put the broth and milk in a medium saucepan and bring to a boil over high heat. Slowly whisk in the polenta, reduce the heat to low, and cook, whisking occasionally, until the polenta is thick and creamy, about 15 minutes. Pull the saucepan from the heat and whisk in the pumpkin, sage with the oil, cheese, salt, and pepper. Hold in a warm place.

For the pork chops: Heat a large skillet over medium-high heat. Dry the pork chops well and season with salt and pepper, to taste. Add the oil to the pan and heat until shimmering. Lay the chops in the pan and cook until lightly browned on the first side, about 4 minutes. Turn the chops and cook until the second side is browned and the chops are cooked through, about 3 minutes more. Transfer the chops to a platter.

Pour off the excess oil from the pan. Add the vinegar, and return the pan to the heat and use a wooden spoon to scrape up any brown bits that cling to the pan. Stir in the mustard, maple syrup, and chicken broth, along with any juices from the pork chops. Cook over high heat until the mixture is syrupy, about 5 minutes. Season to taste.

Serve the pork chops with the pumpkin polenta and drizzle with the sauce.

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