Fill a deep-fryer with the appropriate amount of oil or a large Dutch oven halfway with oil and slowly bring the temperature to 350 degrees F. Line a plate with paper towels and set aside. Dip the mozzarella slices in some milk, then place in some breadcrumbs to coat. Lower the mozzarella into the oil and cook just until the breadcrumbs are golden, about 1 minute. Remove to the paper towel-lined plate and let cool.
In the bowl of a food processor, add the carrot stems, rainbow chard, pine nuts, brown sugar, olive oil, vinegar and smoked sea salt. Pulse until blended well. Place half of the pesto in a small bowl and reserve the remainder for another use.
Dip the tomato slices into the pesto and turn to let the flavors penetrate, then remove from the liquid and let drain.
Place one tomato slice on each of 4 plates, then layer on the fried mozzarella. Layer the remaining tomato slices on top and garnish with shaved Parmesan. Serve chilled.
This recipe was created by a contestant during a cooking competition. It has not been tested for home use.