Prepare the Cucumber Infusion Water: Fill a pitcher with water and add the cucumber slices. Let the cucumber infuse the water and chill in the refrigerator for at least 30 minutes before serving.
Assemble the Sandwich: Lay out the slices of the white bread. Spread mayonnaise on one side of each bread slice. Layer with lettuce, bologna, mortadella, and cheese. Close the sandwiches with the remaining bread slices, mayonnaise side down. Cut in half.
Serve the sandwich with chips, pickle, and a glass of cucumber Infusion water.