In a bowl stir together vinegar and sugar until sugar is dissolved. If using ginger, chop. 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 mixture at least 30 minutes and up to 4 hours.
Mince garlic. Into 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 4 (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.