Step 1

Adjust oven rack to middle position and heat oven to 400 degrees. 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.

Step 2

Process basil, Parmesan, 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.

Step 3

Place 1 mozzarella stick in center of each breast. 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.

Step 4

Heat remaining 2 teaspoons oil in 10-inch ovensafe nonstick skillet over medium-high heat until just smoking. Add breasts, smooth side down, and cook until well browned, about 3 minutes. Flip breasts, transfer skillet to oven, and bake until thickest part of filling registers 160 degrees, 15 to 17 minutes. Transfer breasts to cutting board, tent with aluminum foil, and let rest for 5 minutes.

Step 5

Remove twine, then slice chicken ½ inch thick. Serve.