Even though city ham comes precooked, grilling it is a great way to deepen its smokiness. The glaze here, inspired by Filipino-style barbecue, combines sugary Coca-Cola with pineapple juice, plus soy sauce, ketchup, cider vinegar, and honey, for a complex blend of sweet, tart, salty, and fruity.
To make the glaze: Combine brown sugar, coke, pineapple juice, ketchup, honey, vinegar, and soy sauce in a medium sauce pan. Bring to a boil over medium-high heat, reduce to a simmer, and cook, stirring occasionally until thickened to a syrupy consistency, 20 to 30 minutes. Remove from heat and set aside.
Fire up smoker or grill to 250°F. Wrap ham in heavy duty aluminum foil and place on smoked. Cover and cook until an instant read thermometer registers 100°F when inserted into thickest part of ham, about 2 hours.
Unwrap ham and brush liberally with glaze. Cover and continue to cook for 15 minutes. Brush ham with glaze a second time, cover, and continue to cook until an instant read thermometer registers 120°F when inserted into thickest portion of ham, about 15 minutes more. Remove ham from smoker, let rest for 15 minutes, then serve.
Smoker or grill