Allow the ham to stand at room temperature for 30 minutes to 1 hour before grilling.
Prepare the grill for indirect cooking over medium-low heat (about 325°F).
In a medium saucepan over medium-high heat, melt the butter and cook until it begins to brown, 2 to 4 minutes. Immediately add the remaining glaze ingredients. Lower the heat to a simmer and cook for 5 to 10 minutes, stirring occasionally. Remove the pan from the heat and set aside.
Score the ham in a large crisscross pattern about ½-inch deep on all sides except for the cut side. Place the ham, cut side down, in a large disposable foil pan. Pour the glaze around the ham and into the pan. Cover the ham snugly with aluminum foil, crimping the foil around the rim of the pan. Cook the ham over indirect medium-low heat, with the lid closed, for 1-½ hours. Meanwhile, make the sauce.
In a small nonreactive bowl whisk the sauce ingredients. Cover and refrigerate until ready to serve. (The sauce can be made up to 1 day in advance and refrigerated, covered, until serving time.)
After 1-½ hours of cooking, remove the foil from the ham (but save the foil for later) and quickly spoon some of the glaze over the meat. Continue cooking, with the lid closed, until an instant-read thermometer inserted into the thickest part of the ham (not touching the bone) registers 120°F, 1 to 1-½ hours more, spooning the glaze over the ham every 20 minutes or so. If the glaze gets too dark, cover the ham loosely with the saved aluminum foil for the remainder of the cooking time. Carefully transfer the ham in the pan onto a sheet pan. Tent the ham loosely with foil. Let rest for 15 to 30 minutes.
Cut the ham into thin slices. If desired, drizzle some of the glaze over the slices. Serve the ham warm with the sauce on the side. The ham is also delicious piled high on Hawaiian dinner rolls with some of the sauce spread on one or both sides of the rolls.
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