Start by preheating your oven to 350°F to ensure even cooking throughout. This moderate temperature allows the chicken to cook through while the cheese melts perfectly inside. Having your oven ready prevents any delays once you start the cooking process.
Cut your fresh garlic clove in half to expose the maximum surface area for rubbing. Generously rub the cut garlic over all sides of each chicken breast, pressing firmly to release the oils. This technique infuses the meat with garlic flavor much better than minced garlic would.
Using your sharp knife, carefully slice into each chicken breast about ⅓ inch below the skin. Cut horizontally through the length of the breast, stopping before you cut completely through to the other side. Season both the inside pocket and outside of the chicken generously with salt and pepper.
Heat olive oil in your cast iron skillet over medium-high heat until it shimmers. Place chicken breasts skin-side down and cook for 4-5 minutes until golden brown and crispy. Flip carefully using tongs and cook another 4-5 minutes on the other side, then remove from heat to cool slightly.
Once the chicken is cool enough to handle safely, stuff each pocket with 2-3 tablespoons of shredded mozzarella cheese. Don't overstuff, as the cheese will expand when melting and could burst through the chicken. Press gently to distribute the cheese evenly throughout the pocket.
Layer 2-3 strips of roasted red pepper on top of the cheese in each chicken breast. The peppers add sweetness and vibrant color to contrast with the creamy white cheese. Make sure the peppers sit flat against the cheese for even cooking.
Transfer the skillet to your preheated oven and bake for 8 minutes until the chicken is nearly cooked through. Increase the temperature to 450°F and cook for an additional 5 minutes to crisp the skin and ensure the internal temperature reaches 165°F. Let rest for 3-4 minutes before serving.