Brine ham:

Bring all brining ingredients except ham to a boil in a deep 8-to 9-quart pot (such as a pasta pot), stirring occasionally until sugar and salt are dissolved. Remove from heat and add ham. (Brine should cover ham; if not, make more brine.). Cool mixture 30 minutes, then chill, covered, turning ham once or twice, at least 1day and up to 2.

Prepare grill for cooking using enough hardwood charcoal to fill a charcoal chimney or a shoe box. When coals are ready, push them to one side of grill, leaving about two thirds of grill without coals. Put a drip pan opposite coals.

Make spice rub and cook ham:

Finely grind all spice-rub ingredients together in an electric coffee/spice grinder.

Drain ham and pat dry. Rub ground spices all over ham.

Place ham on gill, fatty side up, over drip pan. Cover grill, leaving cover vents one fourth open and bottom vents fully open, and cook ham, adding a large handful of charcoal every 30 minutes. After 1 hour, turn ham around so other side faces coals and continue to cook until a meat thermometer instead near but not touching bone registers 155F, 1 to 2 hours more.

Make glaze while ham is grilling:

Heat glaze ingredients over moderate heat, stirring, until butter is melted.

During last 30 minutes of grilling, brush ham with glaze several times, letting excess fall into drip pan.

Serve ham:

Transfer cooked ham to a platter and cover loosely with foil. Let stand 30 minutes to 1 hour before carving. Serve with pan drippings (skimmed of fat).

Cooks' notes:

*Gas grill method: preheat grill to moderately low. Place ham, fatty side up, in grill on a rack in a large broiler pan. Close lid, then roast about 3 hours.

*Oven method: Roast ham, fatty side up, on a rack in a large roasting pan in a preheated 350°F oven, about 3 hours.