Process basil, cheese, zest, 2 tablespoons oil, panko, garlic, ¼ teaspoon salt, and ¼ teaspoon pepper in food processor until smooth, about 10 seconds, scraping down sides of bowl as needed. Working with 1 breast at a time and starting at thick end of breast, cut 3½-inch-long horizontal pocket along side, stopping ½ inch from interior edge and 1 inch from tapered bottom. Wiggle tip of knife inside pocket to increase pocket's length to about 4 inches to accommodate cheese. Season breasts inside and out with salt and pepper. Pack 2 tablespoons pesto inside each breast, encasing mozzarella and leaving ¼-inch border along cut edge. Tie each breast with 3 evenly spaced pieces of kitchen twine to enclose filling or use toothpicks to close. Cover with plastic and refrigerate for 3-60 mins. Prepare grill. Grill on med-high until chicken is done.