Special equipment: Six 12-inch metal skewers
Prepare an outdoor grill for medium heat.
Whisk together 1 tablespoon of the 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 six 12-inch metal skewers.
Toast the pine nuts in a small skillet over medium heat, tossing frequently, until light golden, 3 to 4 minutes. Let cool.
Combine the pine nuts, basil, parsley, garlic, 1/2 teaspoon salt and several grinds of pepper in a food processor. Process to make a coarse paste. Add the avocado, lemon juice and remaining 3 tablespoons oil and process until mostly smooth.
Grill the chicken, turning often, until just cooked through, 5 to 7 minutes. Serve the chicken dolloped with the pesto and any extra pesto on the side.
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