For the aioli, blend mayonnaise, basil, arugula, lemon juice, garlic, Parmesan, salt, and pepper in a food processor until completely smooth.
In a frying pan, warm a teaspoon of olive oil on medium heat.
Add the tomatoes and stir to lightly cook and slightly blister their skins. Remove and cool.
In a separate bowl, beat the egg. Dredge the mozzarella in the egg and then in breadcrumbs. Fill a Dutch oven halfway with olive oil, heat until 400 to 450 degrees F, and fry the mozzarella balls approximately 30 seconds each or until golden brown on all sides. Set aside on a paper towel to cool.
Once the mozzarella is at room temperature, use a toothpick or skewer and poke through one tomato then one mozzarella ball, dotting the tops with the basil aioli. Repeat.