In a small bowl, combine cheese, tomatoes, garlic, and basil. Season with pepper and mash together well with a fork.
Butterfly chicken by carefully slicing horizontally along the long edge of each breast, three-quarters of the way through. Open up each breast and spread inside with one-fourth of the feta mixture. Close breast over filling and press edges together to seal. Season lightly with salt and pepper.
In a large ovenproof skillet, heat oil over medium-high heat. Add chicken and cook until browned on both sides, about 2 minutes per side. Transfer skillet to oven and cook until chicken is cooked through, about 20 minutes. Remove from oven and serve warm.