Bring the pork chops to room temperature and season generously with salt and pepper. Heat the oil in a large pan over medium-high heat and sear the chops without moving them until browned on both sides, about 3 minutes on each side (they will not be fully cooked, but still pink in the middle). Transfer the pork chops to a plate, and tent with foil to keep warm.
Reduce the heat to medium-low, and add the shallots and mushrooms to the pan. Sauté for 10 minutes until they are tender and golden brown. Turn the heat back up to medium-high, add the Marsala and thyme, and scrape to release any bits from the bottom of the pan. Add the chicken broth and cream and bring to a simmer, then return the pork chops to the pan, nestling them into the sauce amongst the mushrooms, and continue to simmer until the pork is just cooked through, with a hint of pink in the middle, to an internal temperature of 145°F, about 3 more minutes. The sauce will have reduced and thickened a bit.
Let the chops sit in the sauce for a couple of minutes off of the heat. Serve the chops with the starch of your choice, with the sauce ladled over the top of the pork. Finish with a sprinkle of parsley, if desired.