How to Make It
Heat 1 tablespoon oil in a small skillet over medium. Add 2/3 cup sliced banana peppers, and cook, stirring occasionally, until softened and lightly browned, 5 to 6 minutes. Transfer to bowl of a mini food processor, and let cool completely, about 10 minutes. Add feta and 2 teaspoons lemon juice. Process until thick and creamy, 1 to 2 minutes, stopping to scrape sides of bowl as needed. Season to taste with salt and black pepper. Set aside.
Heat a grill pan over medium. Brush pita with 1/2 tablespoon oil; season to taste with salt and black pepper. Rub chicken breast with 1/2 tablespoon oil, and sprinkle with salt and black pepper. Grill chicken, uncovered, until a meat thermometer inserted in thickest portion registers 155°F, 6 to 8 minutes per side. Remove from pan; let rest 10 minutes. (Internal temperature will continue to rise to 165°F.) Grill pita, uncovered, turning often, until crisp and lightly charred, 4 to 6 minutes. Break pita into 1-inch pieces.
Toss together tomatoes, cucumber, onion, olives, mint, pita pieces, remaining banana peppers, remaining 1 tablespoon oil, and remaining 1 teaspoon lemon juice in a medium bowl. Season to taste with salt and black pepper.
Slice chicken. Spread 1/4 cup banana pepper dip on a serving plate, and top with sliced chicken. Serve with fattoush. Reserve remaining dip for another use.