1. Preheat grill to medium-high heat (350° to 400°). Rinse wood chips in water; drain, and place on a sheet of heavy-duty aluminum foil. Fold foil over, and seal sides to make a packet. Poke several holes in top. Place foil-wrapped chips on grill grate or hot coals.
2. Whisk together vinegar, sugar, celery seed, and 1/2 teaspoon salt in a medium bowl; toss with cucumber ribbons. Set aside.
3. Grill burgers, covered with grill lid, 5 minutes on each side for medium-rare, or until desired degree of doneness. Transfer burgers to a platter, and sprinkle with pepper and remaining 1/2 teaspoon salt. Top each burger with 1 cheese slice; tent loosely with aluminum foil to keep warm.
4. Grill buns 1 minute. Meanwhile, squeeze excess liquid from reserved cucumbers. Place burgers on buns, and top evenly with tomato slices and cucumber relish.