In a blender, combine basil leaves, pine nuts, Parmesan, water, salt, and pepper. Blend until smooth, adding extra water 1 tbsp at a time (up to 2 tbsp) if needed for consistency.
Rub 2 tbsp of pesto onto the chicken breasts, coating them evenly. Season with a pinch of salt and pepper. Let marinate at room temperature for 10 minutes (or refrigerate for up to 2 hours for deeper flavor).
Heat olive oil in a grill pan or skillet over medium-high heat. Cook the chicken breasts for 6-7 minutes per side, pressing gently with a spatula to ensure even browning, until the internal temperature reaches 165°F (74°C). Remove from heat and let rest for 5 minutes before slicing into strips.
Divide lettuce, red onion, avocado, cherry tomatoes, and feta between two plates. Top with sliced chicken and drizzle the reserved pesto over everything.