Tear 4 pieces of aluminum foil that are large enough to encircle each sandwich. Place on a flat work surface.

On top of each foil piece, stack a bottom bun, 1/4 of the ham, a pineapple ring, and 1 slice of cheese. Place the top bun on top. Wrap the sandwich completely with foil. Repeat with remaining sandwiches.

To cook, place the sandwiches on a hot stone near the fire, or over hot coals or a grill rack until hot, about 20 minutes depending upon the temperature of your fire. (Alternatively, you can bake these sandwiches directly on top of the center rack in a 400 degree oven for 15 minutes.)

Let cool slightly, unwrap, and enjoy!