Preheat a grill or grill pan over medium-high heat. Measure 1/3 cup pesto into a separate container. Brush the chicken with the 1/3 cup pesto, then season the chicken with salt and pepper to taste.
Grill the chicken until cooked through and juicy, 2 minutes per side; transfer to a plate. Brush the insides of each roll with olive oil and grill the pieces, oiled side down, until the bread is toasted.
Brush the grilled side of the rolls with the remaining pesto, then top each Panini bottom with one piece grilled chicken and some slices of fresh mozzarella and tomato. Garnish with basil leaves. Serve the sandwiches either open-faced or closed.
To make for a larger group, simply double the recipe!