Put the potatoes and parsnips in a medium pot and cover with water. Bring to a boil over medium heat, salt the water, and cook until tender, 15 to 18 minutes. Drain and put them back into the hot pot. Mash with the milk and salt, pepper, and nutmeg to taste. Add the cheese and stir until melted.
While the potatoes and parsnips come to a boil, rub the chops with lots of salt, pepper, and sage. Heat the extra-virgin olive oil over medium heat in a large oven-safe skillet. Add the garlic and saute for 3 minutes, then remove the garlic with a slotted spoon and raise the heat to medium-high.
Add the chops and brown on both sides, about 5 to 6 minutes. Add the vermouth or wine to the pan and transfer to the oven. Cook the meat through for 7 to 8 minutes more. While the meat cooks, bring a couple of inches of water to a boil over medium heat and salt the water. Add the broccolini and cook for 5 to 6 minutes, then drain. Arrange the chops on 4 plates. Add the butter to the pan and whisk it into the pan juices, then spoon the sauce over the chops. Serve with the mashed potatoes and parsnips and the broccolini alongside.