Toss chicken with half of the pesto; season with salt and pepper. Grill chicken until thoroughly cooked (approximately 6-7 minutes each side). Remove chicken from grill and let cool (about 10-12 minutes).
Once cooled, slice chicken into strips on a cutting board and transfer into a small bowl.
In a large bowl, add lettuce then arrange: red onion, tomato, cucumber, avocado, feta, and egg. Mix in remaining pesto with lemon juice. Taste and adjust seasoning with salt and pepper.
Toss grilled chicken in with the pesto vegetables and serve.
