Combine mayonnaise, vinegar, sugar, and garlic in a medium bowl; season with salt and cayenne. Chill mixture until ready to use.
Carve legs and thighs from chicken. Pull dark meat off bones; coarsely chop meat. Combine dark meat with mayonnaise mixture. Carve off breast portions of chicken. Slice chicken breast meat crosswise on the bias; set aside.
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Preheat grill to high. Brush both side of bread slices with oil. Grill bread, covered, until golden, about 1 minute per side, or simply toast bread.
Divide chicken salad among 4 slices of grilled bread. Top each sandwich with 1/4 cup lettuce, 3 or 4 slices of breast meat, and a second slice of bread. Cut sandwiches in half.
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