In a small bowl whisk together the mayonnaise, horseradish sauce and black pepper until well combined. Set a side.
Spread the unsalted butter on one side of all the bread slices and on the other side spread the mayonnaise mixture (you should have 4 pairs).
Place the butter side of the bread facing down. Put two slices of provolone cheese followed by 4 roast beef slices then two slices of mozzarella cheese on 4 of the slices of bread. Close using the empty slices of bread to create 4 sandwiches with the butter side facing outwards.
Open the lid of the Air Grill and place inside of the basket the Grill plate.
Select the Grill function. Press Temp and select L5, set the time to 5 Minutes. Press Start.
Once the grill finishes preheating and displays Add Food, place inside of the basket the 4 sandwiches pressing them down, close the lid.
When the unit displays Turn Food open the lid and carefully flip the sandwiches, press them down again, close the lid.
Once the cooking time has finished transfer the sandwiches to a cutting board. Carefully remove the top and add some baby arugula, place back the bread slice.