Place a rack in the middle position and preheat the oven to 250°F.
Use 2 tablespoons of the olive oil to grease the inside of a large roasting pan. Place the meat in the pan. Squeeze the juice of the lemon halves evenly over it, then drizzle the remaining 2 tablespoons oil over the meat and rub it in. Season generously with salt and pepper.
Set the roasting pan on the stovetop, across two burners as needed. Sear the meat on high heat, turning it so all sides are deeply browned. (You may need to turn on a vent fan or open a window due to the smoking pan.) Remove the pan from the heat and transfer the meat to a cutting board.
Spread the onions, carrots, and celery evenly in the roasting pan, then return the mutton to the pan, on top of the vegetables, fat side up. Open the oven and pull out the middle rack far enough to place the roasting pan on it, then pour in the broth. Gently slide in the rack, close the oven, and slow-roast the mutton, uncovered, for about 4 hours, or until the meat comes away easily from the bone. During the oven time, check the liquid level and add water as needed, and at times spoon the liquid over the meat.
During the last hour of oven time, transfer the roasting pan briefly to the stovetop. Whisk the flour into the pan drippings and cook over medium heat to thicken and make the gravy. Return the pan to the oven and cook until the meat comes away easily from the bone.
Taste the gravy and add salt and/or pepper as needed. Transfer the meat and vegetables to a platter. Pour the gravy into a bowl and serve hot.
Tip: Halal butcher shops may carry mutton, but be sure it is sheep. Some halal customers who cook Indian food refer to goat meat as mutton.
