Combine all pesto ingredients (except olive oil) into a Vitamix or blender. Turn machine on and off several times, scraping sides with spatula in between. Turn machine on low and slowly add in olive oil, stopping to scrape the sides as needed.
Panini
Spread slices of mozzarella cheese onto a large plate. Sprinkle with salt & pepper.
Meanwhile, to assemble a panini, spread butter on the outside of 2 slices of bread (the sides that will touch the pan). Just use enough butter to prevent from sticking to pan. Spread a spoonful of pesto on the inside of each slice. Then on one slice, add a slice of cheese, a handful of arugula, 2-3 tomato slices, and a second slice of cheese. Close sandwich.
Heat a medium/large pan on medium. Place sandwich on pan and grill for 3-5 minutes on each side. Use a smaller pan’s lid to firmly press sandwich down as it cooks. Alternatively, use a panini press to toast sandwich. Serve once panini is crispy on the outside and the cheese is gooey :)NOTE: I was able to get 3 sandwiches on my pan (so I made 2 batches of paninis), but it all depends on the size of your pan.
Notes
Serve paninis with potatoes, chips or salad. For kids, the paninis might need a couple mins to cool before eating.