Heat the grill over medium-high heat (350-400℉) and oil the grill grates (to do this, dip a wadded paper towel in a little oil and, using tongs, wipe the oil evenly over the grate. Be careful not to use too much oil, because that's a sure-fire way to start a good flare-up—a little goes a long way here.
Cook the chicken over the heat for about 4 minutes per side, until grill marks appear and the chicken is cooked through. The lid should be closed throughout the cooking process.
Reduce the heat to medium-low and spoon about 1 tablespoon of the pesto on top of each chicken breast and spread it across the top into an even layer. Then, evenly sprinkle the tops with the mozzarella cheese. Cover and cook until the cheese is just melted, about 2 minutes.
Remove from the grill and transfer to a clean platter to allow to rest for about 4 minutes.