In a bowl stir together vinegar and sugar until sugar is dissolved. Chop ginger. Slice enough cucumber crosswise into 1/8-inch thick slices to measure 1 1/2 cups and add to vinegar with ginger, tossing well. Marinate at least 30 minutes and up to 4 hours.
Prepare grill or preheat grill pan.
Mince garlic. In a blender tear bread slices and grind into fine crumbs. In a bowl with your hands mix garlic with bread crumbs and remaining burger ingredients until just combined (do not overmix) and form into four 1-inch thick patties. Season burgers with salt and pepper and grill on a lightly oiled rack set 5 to 6-inches over glowing coals about 5 minutes on each side, or until just cooked through. Transfer burgers to buns and top with drained cucumbers.