In a small bowl, whisk together mayonnaise and pesto until blended.

Preheat a grill pan and panini press over medium heat for 5 min. Meanwhile, lay bread slices on a clean work surface. Brush one side of each slice with olive oil, turn the slices over and spread pesto mayonnaise on the slices. Arrange greens on 2 of the slices and top with prosciutto and cheese. Then top each with one of the remaining bread slices, oiled side up.

Remove panini press from grill pan and arrange sandwiches in the pan. Place press on sandwiches and cook until grill marks appear underneath, about 1 min. Turn sandwiches over and cook about 1 min. more. Transfer sandwiches to a cutting board and cut in half. Serve immediately.