In a small bowl, whisk together mayonnaise, 1 tablespoon lemon juice and 1 1/2 teaspoons black pepper. Set aside.
In a separate small bowl, toss arugula with lemon juice and olive oil. Set aside.
Cut the baguette pieces in half lengthwise. Spread all 4 halves with a good helping of lemon pepper spread. To the bottom halves, add a layer of prosciutto, then dressed arugula, and a layer of sliced avocado. Top with shaved Parmesan and other halves of bread.