In a small bowl, mix mayonnaise with salad dressing. Spread the mixture on cut buns. Lubricate each piece of bun with mustard.
Lay out layers of turkey, ham and thinly sliced cheese on each bun. Put chopped cucumbers on top. Close the sandwiches and grease the top and bottom with olive oil.
Heat the pan over medium heat. Place the Cuban sandwich in the pan under the press. Cook for 2 minutes. Remove from heat and cut in half diagonally.