Prepare an outdoor grill or grill pan for medium heat.
Whisk together 1 tbsp oil and the lemon zest in a large bowl. Add the chicken to the oil and toss to coat. Sprinkle with salt and pepper. Divide the chicken and thread onto metal or bamboo skewers.
Toast the pine nuts in a small skillet over medium heat, tossing frequently, until light golden, 3 to 4 minutes.
Combine the pine nuts, basil, parsley, garlic, salt and pepper in a food processor. Process to make a coarse paste. Add the avocado, lemon juice and remaining 3 tbsp oil and process until smooth.
Grill the chicken, turning often, until just cooked through, 5 to 7 minutes. Serve the chicken with the pesto on the side.