Preheat the grill or oven to 400 degrees F. Lay out 4 large squares of foil.
Cut the potatoes into quarters and divide them onto the four pieces of foil. Drizzle each pile of potatoes with 1 teaspoon olive oil and sprinkle with salt and pepper.
Lay one chicken breast over each pile of potatoes. Spread 1 tablespoon of pesto over each chicken breast, and sprinkle with salt and pepper. Fold each piece of foil around the potatoes and chicken, sealing on the top and sides.
Grill or bake the chicken packets for 30-35 minutes. Meanwhile, slice the mozzarella ball into four rounds, and cut four slices of tomato.
Once the chicken is cooked through, carefully open the foil packets and place one slice of mozzarella and one slice of tomato over the top of the chicken. Brush a scant amount of pesto over the top of each tomato. Place the packets, open, over the grill for 5 minutes to let the mozzarella melt.
NOTE: If using larger new potatoes, cut into small bite-size pieces so they cook through.