A beloved American centerpiece dish, this is a true company ham for a special occasion. Everyone loves it because a big ham doesn't come around too often. It's great to see the ham being carved, and the salty, smoky flavor of the meat, combined with the sweetness and crunchy texture of the brown sugar and breadcrumbs, is so satisfying.
Preheat the oven to 325 F. Using a sharp knife, remove the rind from most of the ham, leaving a band around the end of the shank bone and a thin layer of fat all over. Score the fat on top of the ham in a crisscross pattern. Place the ham in a large roasting pan. Pour 3 cups of water into the pan. Bake the ham, uncovered, for 1½–2 hours.
Stir the dark brown sugar and mustard in a medium bowl to make a thick paste. Set aside.
Make breadcrumbs: remove crust from day-old bread and reserve for another use. Cut the bread into large chunks and place into food processor; pulse until crumbly. Heat the olive oil in a large skillet over medium heat. Add the bread crumbs and toast, stirring frequently, until crisp and golden, 5–10 minutes. Season lightly with salt and pepper. Set aside.
Remove the ham from the oven and increase the oven temperature to 350 F. Spread half of the sugar-mustard paste over the scored top of the ham. Bake the ham for 1 more hour.
Remove the ham. Spread the remaining half of the sugar-mustard paste over the ham and pack the breadcrumbs all over the top. Add 1 cup water to the roasting pan if the pan juices have dried out. Return the ham to the oven until the breadcrumbs are deep golden brown, about 15 minutes.
Transfer the ham to a serving platter or a carving board and let it rest, about 10 minutes. To carve, use a large chef's knife and serving fork to cut against the grain, creating long diagonal slices of meat.