Put the ham in a large stockpot, and fill the pot with enough cold water to cover the ham. Bring the water to 170 to 180 degrees (below the boil; if boiled, the ham will crack open at the joint). Cook at that temperature for 3 hours, then let cool in the cooking water.
When ready to proceed with the recipe, remove the ham from the water. With a knife, remove the skin from the ham, and trim the surface of the meat where it is brown and skinlike around the shank bone. To make carving easier later on, remove the pelvis or hip bone by running a knife around it. When the skin has been removed, trim off all but a thin layer of white fat from the surface of the ham.
Run the point of your knife in a crisscross pattern through the remaining fat on the top of the ham, cutting slightly into the surface of the meat. This scoring will help the glaze adhere to the meat while it cooks. Preheat the oven to 350 degrees.
For the peach-mustard glaze: In a small bowl, mix together the preserves, pepper, mustard, balsamic vinegar, and allspice. Spread this coating on the top surface of the ham, place it in a roasting pan, and bake in the 350-degree oven for 30 minutes. Reduce the heat to 325 degrees, and continue cooking the ham for 1 hour. Sprinkle the powdered sugar on top, and cook for another 30 minutes, until nicely glazed. Transfer the ham to a platter, and set it aside in a warm place while you make the peach sauce. Discard the accumulated fat in the roasting pan, and add the 1 cup of water to the drippings in the pan, stirring to loosen and mix in the solidified juices. Add the demi-glace, and bring to a boil. Reserve for use in the peach sauce.
For the peach sauce: Heat the butter in two large skillets, 2 tablespoons of butter per skillet. When it is hot, add half the peach wedges to each skillet, and sauté for 2 to 3 minutes. Sprinkle with the sugar, dividing it between the skillets, and continue cooking for 3 to 4 minutes longer. Add the dried peach julienne and the vinegar, half to each skillet, and stir to mix. Combine the contents of both skillets in one saucepan. Strain the reserved demi-glace mixture over the peaches, and stir in the salt. If the sauce is thinner than you would like, add the dissolved potato starch, and bring to a boil. Boil 1 or 2 minutes. At serving time, stir in the basil.
To serve: To make carving easier, make a vertical cut down into the ham approximately 1 inch above the shank bone. The object is to make a guard that your knife will not go beyond when you slice the ham; this will give a clean bottom edge to the slices and also protect your hand from the knife in case it happens to slide while you are slicing the meat. Slice the ham on the bias, stopping at the cut edge, and arrange the meat on a warm platter. Serve three or four slices of ham per person with a few slices of the peaches and the sauce spooned around and over the meat.