Place all pesto ingredients except olive oil and lemon juice in the bowl of a food processor and pulse until chopped and combined. Add lemon juice and ⅓ cup olive oil and pulse until combined.
Season chicken breast on both sides with salt and pepper. Heat one tablespoon olive oil in a cast-iron skillet over medium heat and sear chicken on both sides until cooked through, about five minutes per side. Remove from heat, allow chicken to cool, and slice.
Toss arugula, tomatoes, cucumber, walnuts, avocado, and chicken in a bowl. Top with pesto.