Using a sharp knife, peel the lemons, removing all the skin and the underlying bitter white pith. The flesh of the fruit should be totally exposed. Over a bowl, cut between the membranes on the sides of each segment to hrefease them. Set aside.

In a large oven-proof sauté pan over medium-high heat, heat 1 tablespoon of the butter and the olive oil. Spread the flour on a plate. Season the chops with salt and pepper and sage, and dredge them in the flour, shaking off any excess. Lay the chops into the sauté pan and cook until nicely browned on both sides, about 5 minutes. Transfer the chops to a platter. If necessary, add another tablespoon of butter to the sauté pan. Add the shallots, garlic, fennel, and carrots. Cook, stirring, until lightly browned, 2 to 3 minutes. Add the sliced mushrooms and cook, stirring, for 6 to 8 minutes. Deglaze the pan with the wine, stirring with a wooden spoon to loosen any browned bits on the bottom of the pan. Add the tomatoes, stock, and demi-glace. Bring to a boil, cover, and transfer to the preheated oven.

Cook for 50 minutes; turn the chops and cook another 30 minutes, or until very tender. Using kitchen tongs or a slotted spoon, place a chop on each plate. Stir the lemon segments into the pan juices, stir in the remaining butter, and spoon the sauce over the chops. Sprinkle with the parsley and serve.